Recent Development in Theories and Numerics: Proceedings of the International Conference on Inverse Problems 9789812383662, 981-238-366-2

Citation preview

Recent Development in

Theories 8~Numerics International Conference on Inverse Problems

This page intentionally left blank

Recent Development in

Theories & Numerics International Conference on Inverse Problems

Hong I 0. For convenience, the function f is supposed to be sufficiently smooth, for example of C'?'. The space below I? is filled with some perfectly reflecting material (a conductor). be filled with a material in such a Let R = ((2 E R3 : 2 3 > f(q)} way that its index of refraction k = w & i is a fixed constant. Here w is the angular frequency. In addition, it is assumed throughout that the index of refraction k satisfies: R e ( k ) > 0 and I m ( k ) 2 0. The case I m ( k ) > 0 accounts for materials which absorb energy. Suppose that a plane wave is incident on r from the top. We then have the following diffraction problem: Given the incident field U I and the periodic stmcture, one wishes to predict the behavior of the outgoing reflected waves. Note that since the medium underneath is a conductor, it does not support any transmitted wave. In the two-dimensional case, there are two fundamental polarizations: T E (transverse electric) and TM (transverse magnetic). In the T E polarization case, i e . , the electric field vector E is assumed to point to the 2 2 axis. In other words, E = u Z 2 , where u = u ( z 1 ,23) is a scalar function. Similarly, in the TM case, the magnetic field H = 2 1 2 2 . For the two-dimensional geometry, the Maxwell equations can be further simplified. Let U I = e i a s l - i ~ z bs e the incident plane wave. Here a = ksin9, p = kcos0, and - 1 ~ 1 2< 9 < r / 2 is the incident angle. From the Maxwell equations (1) and (2), it is straightforward

39

to deduce the following Helmholtz equation:

(A + k2)u = 0 in 0, ulr = 0 ,

(3) (4)

where the homogeneous Dirichlet boundary condition (4)comes from the T E polarization assumption and the assumption that the material is a conductor. Note that for TM polarization, the perfect conductor assumption would imply the homogeneous Neumann boundary condition

Because of the physics, we seek for quasiperiodic solutions to this problem, i.e., the solution u such that ueFiazl is A-periodic for every x3. It is evident that t o completely specify the boundary value problem, we need to impose a radiation condition in the x3 direction. The radiation condition is the boundedness of the scattered fields as x3 tends to infinity. More precisely, we insist that u is composed of bounded outgoing plane waves plus the incident wave U I . Let T be a fixed constant such that T > rnax{f(x~)}. We next present a transparent boundary condition on 2 3 = T which may be derived by a combination of the fundamental solution and the periodicity of the solutions. It allows us t o reduce the scattering problem t o a bounded domain. Let u be the quasiperiodic solution that solves the scattering problem (3) and (4). Then there exists a pseudodifferential operator B of order one 6,26, such that

For the direct scattering problem, questions on existence and uniqueness , are well understood, see for example Chen and Friedman24,Bao6,Dobson26 N6d6lec and Starling35. Basically, the following general result holds. Theorem 1 There is possibly a sequence of frequencies wj with wj + +ca, such that the scattering problem (3), (4), and (5) specified above has a unique quasiperiodic solution provided that w # wj for any j = 1 , 2 , ....

Since k is a fixed constant, for simplicity, we always assume that the direct scattering problem has a unique solution in this paper. In general, because of Theorem 1,this may be arranged by perturbing k or w slightly. Suppose that u (quasiperiodic) solves the scattering problem (3), (4) and (5) for a given incident plane wave U I . The inverse problem can be stated as follows: Determine f(x1) from the knowledge of u(x1,T) or the trace of u.

40

3

Uniqueness for the Inverse Problem

Suppose that for a given incident plane wave U I , u j ( z 1 , z ~() j = 1 , 2 ) is Aquasiperiodic and solve the scattering problem (3), (4), and ( 5 ) with respect to the profiles f j ( z l ) ,where the functions f1 and f 2 are A-periodic. Let T > maz{fl(zl),f~(z:l)} be a fixed constant. Denote h = m a z { f ~ ( z lf2(21)} ), min{f1(z1), fib71)I. We are ready to state a uniqueness result for the inverse problem.

Theorem 2 (Bao 5 , Assume that ~1(z1,T) = that one of the following conditions is satisfied: i). k has a nonzero imaginary part; ii). k is real and h satisfies k2 < 2[h-2 A-2]. Then fl(z1) = fz(z1).

~ 2 ( 2 1 , T ) .Assume

further

+

When k has a nonzero imaginary part, a global uniqueness result was proved by Bao and by Ammari in the biperiodic case. However, in general, global uniqueness may not be possible when k is real. This is evident in the simplest case with a plane wave incident on a flat surface. In this case, the solution of the scattering problem can be written down explicitly. The nonuniqueness is obvious since the scattering fields will remain the same when one moves the flat surface up or down in certain multiples of the wavelength. In the case with real k corresponding to the dielectric medium, one can only prove a local uniqueness theorem. In this case, our uniqueness theorem indicates that any two surface profiles are identical if they generate the same scattering fields (or patterns) and the area in between the two profiles are sufficiently small. Moreover, the smallness of the area is characterized explicitly in terms of a condition which relates the index of refraction k , the period, and the maximum of the difference in height allowed for the two profiles. The proof may be given by an application of Holmgren’s uniqueness theorem and some unique continuation argument. A crucial step is to estimate the first eigenvalue of the Dirichlet Laplacian. In fact, by estimating the eigenvalue, one can get precise idea on how close the profiles need to be for uniqueness t o hold. Global uniqueness of the inverse problem in the dielectric medium case by using a finite number of incident waves have been proved in Bao lo, Hettlich and Kirsch 2 9 . For the 3-D biperiodic problem, a local uniqueness theorem has been obtained by Bao and Zhou l9 where the model and proofs are much more technical. The idea of Bao and Zhou l9 should also yield a local uniqueness result in the TM case.

41

In Kirsch proved a uniqueness theorem by a similar approach as for the general inverse scattering problem in Kirsch and K r e ~ s The ~ ~ .main idea was to prove by using many incident waves the denseness of a set of special solutions. Other related results on inverse diffraction problems may be found in B o r o v i k ~ v ~ ~ , B a o ~ . 4

Stability for the Inverse Problem

In applications, it is impossible to make exact measurements. Thus stability results are crucial in the reconstruction of profiles. This is particularly the case here. In fact, the Rayleigh diffraction theory indicates that the scattered wave may be expressed away from the interface as infinite sum of plane waves, where only a finite number of the plane waves are propagating modes and the rest are exponentially damped. In the far-field, only the propagating modes are detectable. Thus, the measurements are not exact but may be fairly close t o the exact boundary values of the solution. Let first introduce some notations. For any two domains D1 and D2 in R2,denote by d(D1,D2) the Hausdorff distance between them. Denote D = {z; f(z1) < 2 3 < T}, and a sequence of domains Dh = {z; f ( z l ) h o h ( z l ) v ( z l )< z3 < 2’) for any 0 < h < ho, where v(z1) is the normal t o I? = {z3 = f(zl)}. Assume also that the boundary rh = {z3 = f(z1) h o h ( q ) v ( z 1 ) }is periodic of the same period A and is of C2. Further, the function oh satisfies Ioh(z1)I 5 C. Furthermore, for ho is sufficiently small, the sequence of domains is assumed to satisfy that

+ +

Cih I d(D,Dh)F C2h, where C1 and C2 are positive constants. For the fixed incident plane wave u ~assume , that u and u h solve the scattering problem with respect to periodic structures r and r h , respectively. Then we have the following local stability result. Theorem 3

d ( D h , D ) L CIIuhlz,=~ - u I ~ ~ = T I I H ~ / where the constant C may depend on the family {oh}.

~,

(6)

The result indicates that for small h, if the boundary measurements are O ( h ) close to the scattered fields in the If1/’ norm, then Dh is O ( h ) close to D in the Hausdorff distance. The theorem was proved in Bao and Friedman 1 6 . Our proof is based on a variational approach and applications of a unique continuation technique.

42

Actually, in Bao and Friedman16, local Lipschitz type stability results were obtained for a more general class of inverse diffraction problems in both the T E and TM (transverse magnetic) polarizations. More recently, by using the technique of material derivatives with respect t o the variation of the dielectric coefficient, Elschner and Schmidt have generalized the local stability result to the case of polygonal (grating) interfaces. A global stability result has been obtained by Bruckner, Cheng, and Yamamoto 23 under certain additional assumptions equivalent to the validness of the maximum principle. The stability question becomes much more challenging in the 3-D biperiodic case. Until now, the only result available is a local stability result similar to Theorem 3 proved by Bao and Zhou 19. Finally, we mention that local stability results for other inverse problems, for example, inverse conductivity problems, were previously obtained in Bellout and Friedman2’, Bellout et al 21. 5

Optimal Design

Given the incident field, the optimal design problem concerns the creation of grating profiles that give rise to some specified diffraction patterns. The problem can be posed as a nonlinear least-squares problem. Difficulties arise since the scattering pattern depends on the interface in a very implicit fashion and in general the set over which the function is minimized is neither convex nor closed. The formulation of the design problem is very close to similar problems in elasticity, for which fast and efficient algorithms have recently been developed. Initial progress on the design problem has been made via weak convergence analysis methods by Achdou and Pironneau 2 , Dobson 2 6 , and the homogenization theory by Bao and Bonnetier l1 along with the “relaxation” technique of Kohn and Strang 31. The main idea is to allow the grating profiles to be highly oscillating and to use relaxed formulation of the optimization problem. The crucial step is to determine the relaxed formulation which involves materials and the effective dielectric properties l l . We refer to Bao et al for additional results on this and related design problems. Another important direction in optimal design of diffractive optics is to design resonance^^^. One of the most exciting new developments in diffractive optics involves the integration of a zero-order grating with a planar waveguide t o create a resonance. Such structures, known as guided-mode resonance filters, have been demonstrated to yield ultra-narrow bandwidth filters for a selected center wavelength and polarization with w 100% reflectance 34. With such extraordinary potential performance, these “resonant reflectors” have attracted attention for many applications, such as lossless spectral fill 4 > l 5 7 l 7

43

ters with arbitrarily narrow, controllable linewidth, efficient and low-power optical switch elements, 100% reflective narrow-band spectrally selective mirrors, polarization control, high-precision sensors, lasers, and integrated optics. Significant recent progress has been made in Huang 30 for solving an interesting optimal design problem: t o determine the structure and the material that give rise t o a resonance at some specified wavelength. By using the variational approach, the design process may be formulated as an optimization problem where the diffraction grating and waveguide problems are solved repeatedly.

6

Future Directions

A closely related problem is t o determine the periodic (grating) structure ruled on some nonconductive optical material. In this situation, one places optical detectors both above and below the material. The measurements consist of information on the reflected wave and transmitted wave. In the TE case, the model equation takes the same form as Equation (3). However, the boundary condition (4)is no longer valid. Instead, the direct problem may be formulated in a “box” with nonlocal boundary conditions that are similar to (5) on the top and at the bottom. We believe that a local uniqueness theorem for this inverse problem may be proved by modifying the proof of Theorem 3.1. A local stability result was established in Bao and Friedman16. No result is available in the biperiodic case. Another interesting problem concerns global uniqueness for the inverse diffraction problems. In particular, no result is available in the TM (transverse magnetic) case or the biperiodic case. The corresponding inverse problem turns out t o be much more difficult. It is not clear whether additional data such as a finite number of incident waves would be sufficient to assure global uniqueness. The difficulty lies in the fact that the first eigenvalue of the Neumann or vector Laplacian does not have the monotone property with respect to the domain or the diameter of the domain. So far, we were only able t o prove some local stability results l6 by combining a variational approach and the analytic index theory. Numerical solution of the design and inverse diffraction problems is of great interest. As one might expect from the local uniqueness and stability results reported here that some a priori knowledge is necessary in order to determine the structure. An ongoing research is to restrict one’s attention t o a class of curves with certain geometry and then solve the inverse problem by an optimization method. A significant future direction is to study the inverse and design problems in nonlinear optics. It has been observed that the use of gratings can

44

significantly enhance the nonlinear effects of second harmonic generation in nonlinear optics. The field is widely open. We refer the reader to Bao, Huang, and Schmidt for some references and preliminary results on optimal design of nonlinear gratings. Acknowledgments

The research of the author was partially supported by the NSF Applied Mathematics Programs grant DMS 0104001, the NSF Western Europe Programs grant I N T 98-15798, the Office of Naval Research (ONR) grant N000140210365, and an Intramural Research Grants Program grant of Michigan State University. References

1. T. Abboud, Electromagnetic waves in periodic media, in Second International Conference on Mathematics and Numerical Aspects of Wave Propagation, ed. R. Kleinman et al.( SIAM, Philadelphia , 1-9(1993)). 2. Y. Achdou and 0. Pironneau, Optimization of a photocell, Optimal Control Appl. Meth. 12 , 221-246(1991). 3. H. Ammari, Uniqueness theorems f o r an inverse problem in a doubly periodic structure, Inverse Problems 11 , 823-833( 1995). 4. G. Bao, A uniqueness theorem f o r an inverse problem in periodic diffractive optics, Inverse Problems 10, 335-340 (1994). 5. G. Bao, An inverse diffraction problem in periodic structures, in Proceedings of Third International Conference o n Mathematical and Numerical Aspects of Wave Propagation, Ed. by G . Cohen( SIAM, Philadelphia, 694-704( 1995)). 6. G. Bao, Finite elements approximation of time harmonic waves in periodic structures, SIAM J. Numer. Anal. 32,1155-1169(1995). 7. G. Bao, Numerical analysis of diffraction b y periodic structures: T M polarization, Numer. Math. 75, 1-16 (1996). 8. G. Bao, Variational approximation of Maxwell’s equations in biperiodic structures, SIAM J. Appl. Math. 57, 364-381(1997). 9. G. Bao, O n the relation between the coeficients and solutions f o r a diffraction problem, Inverse Problems 14 , 787-798 (1998). 10. G. Bao, Inverse diffraction b y a periodic perfect conductor with several measurements, in Inverse Problems in Engineering, Theory and Practice, Ed. D. Delaunay, Y. Jarny, and K.A. Woodbury( ASME, 297-303,1998).

45

11. G. Bao and E. Bonnetier, Optimal design of periodic diffractive structures, Appl. Math. Optim. 43 , 103-116 (2001). 12. G. Bao, L. Cowsar, and W. Masters, ed., Mathematical Modeling in Optical Science, the SIAM Frontiers in Applied Mathematics, SIAM, Philadelphia (2001). 13. G. Bao and D. Dobson, O n the scattering by biperiodic structures, Proc. Am. Math. SOC.1 2 8 , 2715-2723(2000). 14. G. Bao, D. Dobson, and J. A. Cox, Mathematical studies of rigorous grating theory, J. Opt. SOC.Am. A 1 2 , 1029-1042(1995). 15. G. Bao, D. Dobson, and K. Ramdani, A constraint on the maximum reflectance of rapidly oscillating dielectric gratings, SIAM J. Control. Opt. 4 0 , 1858-1866(2002). 16. G. Bao and A. Friedman, Inverse problems for scattering by periodic structures, Arch. Rat. Mech. Anal. 132 , 49-72(1995). 17. G. Bao, K. Huang, and G. Schmidt, Optimal design of nonlinear gratings, submitted. 18. G. Bao and H. Yang, A least-squares finite element analysis for diffraction problems, SIAM J. Numer. Anal. 2 , 665-682(2000). 19. G. Bao and Z. Zhou, A n inverse problem for scattering b y a doubly periodic structure, Trans. Ameri. Math. SOC.350 , 4089-4103(1998). 20. H. Bellout and A. Friedman, Identification problems in potential theory, Arch. Rational Mech. Anal. 101 , 143-160(1988). 21. H. Bellout, A. Friedman, and V. Isakov, Stability for an inverse problem in potential theory, Tran. Amer. Math. SOC.332 , 271-296(1992). 22. I. Borovikov, Uniqueness of solutions to one inverse diffraction problem, Differentsial’nye Uravneniya 28 , 827-831( 1992). 23. G. Bruckner, J. Cheng, and M. Yamamoto, An inverse problem in diffractive optics: conditional stability, Inverse Problems 18 , 415-433(2002). 24. X. Chen and A. Friedman, Maxwell’s equations in a periodic structure, Trans. Amer. Math. SOC.323 , 465-507(1991). 25. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, New York, 1992). 26. D. Dobson, Optimal design of periodic antireflective structures for the Helmholtz equation, Euro. J. Appl. Math. 4 , 321-340(1993). 27. J. Elschner and G. Schmidt, Numerical solution of optimal design problems for binary gratings, J. Comput. Phys. 146 , 603-626(1998). 28. J. Elschner and G. Schmidt, Inverse scattering for periodic structures: stability of polygonal interfaces, Inverse Problems 17 , 1817-1829(2001). 29. F. Hettlich and A. Kirsch, Schiffer’s theorem in inverse scattering theory for periodic structures, Inverse Problems 13 , 351-361( 1997).

46

30. K. Huang, Optimal Design of Diffractive Optics, Ph.D. Thesis ( Michigan State Univ., 2002). 31. R. Kohn and G. Strang, Optimal design and relaxation of variational problems 1, 11, III, Comm. Pure Appl. Math. 39 , 113-137, 139-182, 353-377 (1986). 32. A. Kirsch, Uniqueness theorems in inverse scattering theory for periodic structures, Inverse Problems 10 , 145-152( 1994). 33. A. Kirsch and R. Kress, Uniqueness in inverse scattering, Inverse Problems 9 , 285-299(1993). 34. R. Magnusson and S. Wang, New principle for optical filters, Appl. Phys. Lett. 61 , 1022-1024(1992). 35. J. C. Nkdklec and F. Starling, Integral equation methods in a quasiperiodic diffraction problem for the time-harmonic Maxwell’s equations, SIAM J. Math. Anal. 2 2 , 1679-1701(1991). 36. R. Petit, Electromagnetic Theory of Gratings, inTopics in Current Physics, Vol. 22’ ed.R. Petit( Springer-Verlag, Heidelberg, 1980).

THE INVERSE PROBLEM OF OPTION PRICING VICTOR ISAKOV Department of Mathematics and Statistics, Wichita State University, Wichita, KS 67260-0033, U . S . A . E - m a i h i c t o r . [email protected] We consider the problem of recovery of the volatility coefficient of the Black-Scholes equation for option prices as functions of time and of stock price. We give most recent results about uniqueness and stability of reconstruction of volatility from market data and discuss relations with stochastic partial differential equations. We suggest two algorithms of numerical reconstruction, using a parametrix and the linearized inverse problem. We give the results of some numerical tests. For simplicity, we handle only European options.

1

The Black-Schools Equation

For any stock price, 0 < s < co,and time , 0 < t < T , a price u for an option expiring at time T satisfies the following partial differential equation du

-

dt

+ -s21

a2U

CJ

(s)-

dS2

+ sp-dU dS

- TU = 0

Here, ~ ( s is) the volatility coefficient that satisfies 0 < m < ~ ( s 1 and CT sufficiently large, for a precise statement see Theorem 3.1. This still describes an idealized situation: in practice one never aims at solving (7) precisely, one rather chooses a from a sequence of test parameters and determines a~ E {an = qnagJ n = 0,1,2, ...}, for a fixed 0 < q < 1 by requiring IIAxtN,A,

-

y611

5 r6 +

+

~ I A X -~ y61l ~ > , r~ b ~

(8) CTE

for n

2/q, > 911x+11/4q, then

11xt,l\,- x+Il

=

o(62u’(2u+1) 1 .

The above theorem shows that we can e.g. choose p = q = 1/2 and still obtain optimal convergence rates. Such a choice is preferable for large values of Q which is the case in the beginning of our iterative search for the optimal regularization parameter. Optimal convergence rates cannot be achieved in general if p q < 1.

+

4

Wavelet Galerkin methods for operator equations

In recent years, much effort has been spent to design efficient numerical schemes based on wavelets. The most far-reaching results were obtained for

62

operator equations of the form

du=

(12)

f6,

where d : H + H' is a linear operator from a Hilbert space H into its normed dual H'. In our applications, H will typically be a Sobolev space H t on some domain R C Rd or on a closed manifold. We assume that d is boundedly invertible so that lIdUIIHJ

(13)

11V11H,

holds. This setting fits perfectly to the normal equation (5) arising in the inner iteration, i.e., to the problem

+

zc6, = ( A * A a1)-lA*y6 ,

(14)

+

since, as already stated above, A = ( A * A a1) is boundedly invertible on Lz(R). More precisely, the operator norm of d-l is bounded by lId-'\l 5 a-l. The right hand side f6 = A*y6 satisfies 11 fs - f 11 5 llA*ll 6. Before we discuss later on the specific problems arising in the numerical treatment of (14), let us briefly recall the basic numerical concepts. We are especially interested in adaptive schemes, and we shall focus on numerical algorithms based on wavelets, i.e., the basis functions are taken from a family @ = {+A, X E J } satisfying the following fundamental assumptions: 0

@ induces n o r m equivalences for a whole scale of Sobolev spaces,

11 CACJ~A'$A~~H~

( C ~ ~ ~ 2 ~ ' ~ ' ~ l SOd 5 A S1 5~ S) l ;~ ' ~ ,

possesses the cancellation property I(v,+

~ ) l5

2-~A~m~vl~m~.,pp~

0

+A

0

the wavelets are local in the sense that diam(supp+A)

-

2-1'1, X E J .

Nowadays, several constructions of bases satisfying these assumptions are available Our goal is to develop a suitable Galerkin scheme to approximate the solution of (14). Therefore we consider subspaces of the form 4,77879.

SA := {+A

: X E

A},

A

C

J,

(15)

and project our problem onto these spaces, i.e., the Galerkin approximation is defined by

UA

( d u i , v )= (f6,v) ,

v E SA.

(16)

In an adaptive scheme, the goal is always to find a possibly small set A C J such that the actual error is below some given tolerance. In principle, such a scheme consists of the following three steps:

63 0

compute the current Galerkin approximation U A ;

0

estimate the error 11u6 - uill in some suitable norm, with u6 = A-'f6;

0

add wavelets if necessary which yields a new index set

A.

For the second step, one clearly needs an a posteriori error estimator since the exact solution u is unknown, and for the third step one has to develop a suitable refinement strategy so that the whole algorithm converges. In the wavelet setting, an error estimator can be easily constructed by employing assumption (13), norm equivalences, and Galerkin orthogonality: 6

7 11u - uA)1 5 ) ) u- u 6 )+ ) 11u6 where the first term is controlled by the Tikhonov regularization and the second term gives rise to the error estimator via

IIu

6 -

6 UAllHt

- llf6

llA(u6- u i ) l l H - t =

-

2-2tlxlI(rA,?bA)12

IITAIIH-t

L

(17)

duillH-t

)

1'2.

A

In our example for the inverse heat problem we have A : &(a)4 &(a), i.e. = 0. F'rom (17), we observe that the current error can be estimated by computing the wavelet coefficients of the residual r A = f6 - Aui. Intuitively, the residual weights p x := 2 - t l x l I ( r ~+x)l , serve as local error indicators. Therefore a suitable refinement strategy can be derived by adding those wavelets which produce large entries in the expansion of the residual, i.e., we define the new index set in such a way that

t

A

for some suitable parameter p. However, this strategy is not directly numerically realizable since catching the bulk of the residual requires knowing all its wavelet coefficients. Nevertheless, in 6 , it was shown that a judicious variant of this idea exploiting the cancellation property of wavelets indeed leads to an implementable and convergent algorithm, i.e., given a tolerance E , the adaptive scheme produces a final index set i E such that 6

IIU

6

- UA,II

5E

(19)

by using only information on the given data. Moreover, in 5 , subtle generalizations have been derived which yield asymptotically optimal schemes in the

64

sense that (within a certain range) the convergence rate of best N-term approximation is achieved at a computational expense which stays proportional to the number N = of degrees of freedom. Furthermore, in ', a first efficient numerical realization is documented. As already stated above, we suggest to use this strategy for the numerical treatment of the basic problem (14),

Inel

~t = (A'A + a.I)-'A*y6 .

(20)

Clearly this problem fits perfectly into the framework described above. However, as explained in detail in the design of an implementable refinement strategy requires some compressibility properties of the underlying operator. For the special operators considered here, this issue will be further analyzed in the near future. Moreover, for an efficient implementation, the problem remains how to compute the entries of the associated stiffness matrix 576,

( - h ) x , x r := ( W x ,+A) = (A$xt, A+x)

+ Q(+AJ,

+A)

(21)

and of the right-hand side (A*y6)x = (y6,A+x).

(22)

Fortunately, the adjoint operator A* is not needed, but nevertheless the task is nontrivial since the operator A is induced by the forward problem (4), i.e., it is given as a parabolic equation. We intend to solve this problem with another fully adaptive scheme as we shall now explain. Following the basic investigations in we treat our parabolic equation as an abstract Cauchy problem 213,

u'(tj

+ Bu(tj = 0 ,

t E (0,T],

u(0)= uo. Usually, this problem is treated by the method of lines. Discretization in space first leads to a block system of ordinary differential equations. However, as already outlined in for an adaptive approach the other discretization sequence, first time then space, which is classically known as the method of Rothe 24 seems to be preferable. Then (23) is viewed as an ordinary differential equation in some suitable Hilbert space which, due to stability reasons, is solved by an implicit scheme with time-step control. Then, in each step, a certain elliptic subproblem has to be solved. However, since these subproblems are boundedly invertible in the sense of (13), they can again be efficiently discretized by employing the well-known adaptive wavelet algorithm. Clearly, the convergence and efficiency of this strategy has to be analyzed in detail. This will be performed in the near future. 233,

65

Acknowledgments This research was partially supported by the Deutsche Forschungsgemeinschaft (DFG), Grant Da360/4-1 and the Bundesministerium fur Bildung, Wissenschaft, Forschung und Technologie under grant number BMBF-03MSMlHB.

References 1. A. Barinka, T. Barsch, P. Charton, A. Cohen, S. Dahlke, W. Dahmen, and K. Urban, Adaptive wavelet schemes for elliptic problems - Implementation and numerical experiments, SIAM J. Scientzfic Comp. 23(3), 910-939 (2001). 2. F. Bornemann, An adaptive multilevel approach to parabolic equations I. General theory and 1D implementations, Impact Comput. Sci. Engrg. 2, 279-317 (1990). 3. F. Bornemann, An adaptive multilevel approach to parabolic equations 11. Variableorder time discretization based on a multiplicative error correction, Impact Comput. Sci. Engrg. 3, 93-122 (1991). 4. C. Canuto, A. Tabacco, and K. Urban, The wavelet element method, part 11: Realization and additional features in 2d and 3d, Appl. Comp. H a m . Anal. 8 , 123-165 (2000). 5. A. Cohen, W. Dahmen, and R. DeVore, Adaptive wavelet methods for elliptic operator equations - Convergence rates, Math. Comp. 70,22-75 (2001). 6. S. Dahlke, W. Dahmen, R. Hochmuth, and R. Schneider, Stable multiscale bases and local error estimation for elliptic problems, Appl. Numer. Math. 23, 21-47 (1997). 7. W. Dahmen and R. Schneider, Composite wavelet bases for operator equations, Math. Comput. 68, 1533-1567 (1999). 8. W. Dahmen and R. Schneider, Wavelets on manifolds I: Construction and domain decomposition, SIAM J. Math. Anal. 31, 184-230 (1999). 9. W. Dahmen and R. Schneider, Wavelets with complementary boundary conditions - function spaces on the cube, Res. in Math. 34, 255-293 (1998). 10. V. Dicken and P. Maafi, Wavelet-Galerkin methods for ill-posed problems, J. Inw. and Ill-posed Probl. 4(3), 203-222 (1996). 11. H.W. Engl, Discrepancy principles for Tikhonov regularization of illposed problems leading to optimal convergence rates, J. Opti. Theory Appl. 52, 209-215 (1987).

66

12. H.W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer, Boston, (1996). 13. H. Gfrerer, An a posteriori parameter choice for ordinary and iterated Tikhonov regularization of ill-posed problems leading to optimal convergence rates, Math. Comp. 49, 507-522 (1987). 14. C.W. Groetsch, The Theory of Tikhonov Regularization for Redholm Equations of the First Kind, Pitman, Boston (1984). 15. J.T. King and A. Neubauer, A variant of finite-dimensional Tikhonov regularization with a-posteriori parameter choice, Computing 40, 91-109 (1988). 16. A.K. Louis, Inverse und schlechtgestellte Probleme, Teubner, Stuttgart (1989). 17. A.K. Louis, P. Maan, and A. Rieder, Wavelets - Theorie und Anwendungen, Teubner, Stuttgart (1994). English version: Wiley, Chichester. 18. P. Maafi and R. Ramlau, Wavelet accelerated regularization methods for hyperthermia treatment planning, Int. J. Imag. Sys. and Tech., 7, 191-199 (1996). 19. P. Maan and A. Rieder, Wavelet-accelerated Tikhonov-regularisation with applications, in “Inverse Problems in Medical Imaging and Nondestructive Testing”, eds. H.W. Engl, A.K. Louis, and W. Rundell, Springer, Wien, New York, pp. 134-159 (1997). 20. A. Neubauer, An a posteriori parameter selection choice for Tikhonov regularization in Hilbert scales leading to optimal convergence rates, SIAM J. Numer. Anal. 25, 1313-1326 (1988). 21. A. Neubauer, An a posteriori parameter choice for Tikhonov regularization in the presence of modeling error, Appl. Num. Math. 4, 507-519 (1988). 22. S.V. Pereverzev, Optimization of projection methods for solving ill-posed problems, Computing 55 (1995). 23. J. Reinhardt, On a sideways parabolic equation, Inverse Problems 13, 297-309 (1997). 24. E. Rothe, Zweidimensionale parabolische Randwertaufgabe als Grenzfall eindimensionaler Randwertaufgaben, Math. Ann. 102, 650-670 (1930). 25. M. Yamamoto and J. Zou, Simultaneous reconstruction of the initial temperature and heat radiation coefficient, Inverse Problems 17, 11811202 (2001).

ESTIMATION OF DISCONTINUOUS SOLUTIONS OF ILL-POSED PROBLEMS BY REGULARIZATION FOR SURFACE REPRESENTATIONS: NUMERICAL REALIZATION VIA MOVING GRIDS ANDREAS NEUBAUER Instatut fiir Industrie mathematik, Johannes-Kepler- Universitat, A-40.40 Linz, Austria E-mail: [email protected] In this paper we discuss the numerical realization of a new regularization method, regularization for surface representations, which is well-suited for ill-posed problems with discontinuous solutions: this realization is essentially based on moving grids. After describing the method we present several numerical examples showing that this combination with moving grids is a powerful tool to identify discontinuities in two-dimensional problems.

1

Introduction

In this paper we study the estimation of discontinuous solutions of linear or nonlinear ill-posed problems

F(f)= 9

(1)

from noisy measurements g6 of g satisfying ([g' - 911 5 6, where

F : D ( F ) ( C X ) -+ Y and X and Y are Hilbert spaces. Tikhonov regularization is well known to stabilize ill-posed problems 3. In this method an exact solution of (1) is approximated by a minimizer of the functional

ft

IIF(f)- g6Il2+ crp(f - f*) (2) where f * is an initial guess of the exact solution and p ( . ) is a properly chosen penalty term. Usually one uses the penalty term 7

P ( f - f*) =

Ilf - f*1I2.

(3) However, this is not appropriate for ill-posed problems with discontinuous solutions, since it has a smoothing effect on the regularized solutions. Using the bounded variation norm in (2) as penalty term has turned out to be an effective regularization method lJJo. A major drawback of this approach, however, is that this norm is not differentiable. 67

68

Neubauer and Scherzer introduced a new approach for regularizing problems with discontinuous solutions, regularization f o r curve representations. The essence of this method is to replace a discontinuous function by its continuous graph. This allows a combination with the usual Tikhonov regularization in Hilbert spaces ((2), (3)) and, therefore, all the results about convergence from the general theory on nonlinear Tikhonov regularization are applicable. The method was successfully applied to one-dimensional parameter estimation problems by Kindermann and Neubauer 5 . Kindermann and Neubauer generalized the method to two-dimensional problems, regularization f o r surface representations, and applied it to the linear problem of deblurring images. The idea of this method is as follows: Let f be a C1-function, then its graph Gf := { (z, y , f(z,y ) ) E R3} defines a surface in R 3 .Let ( a ( u ,w),b(u,w),c ( u ,v)) be an equivalent parameterization of G f , then f can be recovered by f ( z , y ) = c ( ( a , b ) - ' ( z , y ) ) . It was shown that discontinuous functions f that are in a certain subset of the functions of bounded variation may be parameterized by smooth parameterizations. To guarantee that the parameterized surface may be interpreted as the graph of a function, we restricted ourselves to parameterizations

a ( u , v ) = a(u)

b(u,w) = b(w)

(4)

satisfying ( a ,b, c) E D with

D

:= { ( a ,b, c ) E

x

:

a(1)= 1 = b(l), 2

o a.e., i, 2 o a.e.1 ,

x := { ( a , b , c )E H1[0,1]x H 1 [ 0 , 1 ]x H,1(R)

:

a(0) = 0 = b ( O ) } .

(5)

This method, which was successfully applied to two-dimensional parameter estimation problems , is still quite restrictive: due to the special choice of surface parameterizations, discontinuities in solutions f = c(a-' (z), b-l ( y ) ) are only allowed to occur on lines parallel either to the z- or the y-axis. The general case of parameterizations was treated from a theoretical point of view in the PhD-Thesis of Stefan Kindermann '. He showed that a general parameterization, f ( ( a ( u ,w), b ( u ,w)) = c(u,v), exists for all functions of bounded variation with compact support. He also gave conditions on F that guarantee that the Tikhonov regularized solutions converge to the exact one. Let us consider the two-dimensional linear integral equation

where R = [0,1l2 c R2 is the unit square and k E L2(R2). Note that F : L2(R) -+ L2(R) is compact. The reformulation of this equation in terms

69

of ( a ,b, c) yields

(7)

with

J ( a ,b)(u,v) = au(u,v)b,(u,).

-

a,(u,v)bu(u,). .

This is now a nonlinear integral equation with respect t o a and b. G is at least well defined for those a , b, c E H1(0)that are parameterizations of functions of bounded variation with compact support 4 . This problem is stabilized via nonlinear Tikhonov regularization, where we can use the standard penalty term, i.e., we are looking for minimizers (a:, b:, c): of the functional

For (a,b) : R + R t o be admissible in the sense that the parameterized surface may be interpreted as the graph of a function, it is necessary that ( a ,b ) is one t o one and onto. Besides some boundary conditions this means that J ( a , b) is not allowed t o change sign (except for a set of measure zero). For the special parameterization (4) this was guaranteed through the conditions in (5). These conditions were also easy to check in the numerical realization. If, in the general case, we used bilinear finite element functions for the approximation of a , b, and c, (8) would turn into a nonlinear minimization problem with respect to the coefficients of a , b, and c governed by nonlinear constraints in each node of the finite element grid of R. This is much too involved. Therefore, we will present a numerical realization based on moving grids which is much faster and yields excellent results. The method of moving grids is described in the next section. In Section 3 we combine it with regularization for surface representations. Finally, numerical results are presented in Section 4. 2

Moving Grids: The Deformation Method

Moving grids have been developed for the numerical solution of time dependent partial differential equations. A drawback of using fixed grids occurs when the solution of the PDE exhibits large variations due to, for example, shock waves or moving fronts. Due to its static feature the fixed grid is unable to efficiently and accurately resolve such variations. One can improve this by using adaptive grids. The idea is t o generate the grids such that nodes will

70

be concentrated in regions where the solution changes rapidly and fewer grid points are used in regions where negligible changes occur. There are essentially two strategies for grid adaption: local refinement and moving grids. In local refinement nodes are inserted where and when they are needed. This method is flexible and easy to conform the boundary. However, the solver and data structure have to be modified after insertion or deletion of nodes. For moving grids the total number of nodes and the connectivity between them is fixed. The nodes are redistributed where and when they are needed. There are different methods to move the nodes around. We only describe the so called traditional deformation method, which is based on a result from differential geometry. It provides direct control over the cell size of the adaptive grid and the node velocities are directly determined. It turned out that it is also efficient for the numerical calculation of discontinuous solutions of ill-posed problems in combination with regularization for surface representations. The description follows the lines in Liao et al *. Suppose that u satisfies Ut

= Jqu),

(9)

where L is a differential operator defined on a physical domain R c R2. The idea is now to construct a transformation q5 : fi x [O,T]+ R which moves a fixed number of grid points on R to adapt to the numerical solution. Of course q5 must be one to one and onto. It is well known that if the Jacobian determinant of a transformation q5 is positive in fl, then 4 is one to one in all of fi. This ensures that the grid will not fold onto itself. Therefore, the deformation method constructs q5 such that detVq5 = m(q5,t), where m is a positive monitor function. This assures precise control over the cell size relative to the fixed initial grid. Suppose that the solution of (9) has been computed at time step t = t k - 1 and a preliminary computation has been done at time level t = t k . Assume that we have some positive error estimator e(q, t ) at the time steps t k - 1 and t k . Let us define the monitor function

where y ( t ) is a positive scaling parameter so that

s,

1 m(ll,dq =

holds. Note that m is small in regions where the error is large and large in regions where the error is small. We then seek a transformation

71

I$ : fi

X [tk-l,tk]

+ 0 such that

< fi , < E fi

detV+( 0.

The eigenvalues of ( K - I ) are given by X1,2 = ( k l l + k z z - 2 ) f ~ ( 2k i i - k z ~ ) ~ + 4 k ~ , Note that since K # I we have that A 1 # A2 and that an eigenvalue can be zero if and only if k11k22 - kf2 = kll k22 - 1. In particular in the

+

130

orthotropic case, i.e. k12 = 0, A1,2 = ( k 1 1 + k 2 2 - 2 )2* t k 1 1 - k 2 2 ~ can be zero if and only if k11 k22 = 1 kllk22, i.e. when Icll = 1 or k22 = 1. So we have the possibility to choose better experiments for an orthotropic inclusion for which

+

k l l # 1 and

+

k22

#

1. Let us therefore consider K =

[

:].

Then

A1

= 1 and

A2

= 2 with the corresponding normalized eigenvectors for ( K - I) given by

a,

=

[i]

and

or f2(2) =

a2 =

a2 0 s

[ ].

Then we can experiment with fl(3) = a,

= y, which give TI =

0

3 = z,

s,"" cos(O)g(cos(O),sin(8))dO and

2n

T2 = Jo sin(Q)g(cos(O),sin(O))dO, respectively. From the estimates (27) we can choose the experiment. However, in general there is no control between g and D . All one can do in practice is to calculate TI and T2 and according t o (27) decide which experiment is the better, i.e. which estimates are the sharper.

4

Boundary Integral Formulation

The refraction model under investigation, given by Eqs. (1) and (2) and the corresponding transmission conditions, can be recast in the more convenient form by defining 4 = 41 in R - D and 4 = 4 2 in D, where 41 and 42 satisfy V2& = 0,

in

V (KV42) = 0,

41 = 4 2 ,

-D

(28)

in D

(30)

a41 "2 -= -(KV42) o n + = -dn-

dn* '

on a~

where E* = Kn+. Let us assume now that R and D are simply connected and have C2 boundary. Then $1 E H2(R- D) n C(n) and 42 E H2 (D ), see Ladyzenskaia4 (p.198), and thus Green's formula is applicable. Prior t o this study, Kang and Seo8 and Ki and Sheeng developed an integral representation for the isotropic case. However, their approach cannot be extended to the anisotropic case so easily. Instead, the boundary integral methods of Duraiswami et al. lo and Lesnicl' for the isotropic case can be extended to the anisotropic case as follows. Let G and G K be fundamental solutions of the Laplace equation and anisotropic Laplace Eq.(l) in R d ,

131

respectively,

-wZ 2K n ( R ) , d = 2 G K (a,$1 =

w,

(33) d=3

where T =I g - 5 1, I K-' I is the determinant of the inverse matrix K-' and the geodesic distance R is defined by R2 = K - l ( g - 5) (a: - f ) . Considering, for simplicity, the Dirichlet boundary condition in (29), i.e. &(x) = f(x) for 2 E dR, and applying the interface transmission condition (31) we obtain the following integral representation formulae:

where ~ ( 2=)1, if r: E R - d D and ~ ( g=)0.5 if g E dR U d D . Analytical solutions t o (34) and (35) are in general not feasible and therefore some form of numerical approximation given by the boundary element method, see Chang et al 12, has t o be performed. In two-dimensions, i.e. d = 2, the boundary dR is discretized uniformly into M constant boundary elements in a counterclockwise sense, and the boundary d D into M constant boundary elements in both a counterclockwise and clockwise sense. Then applying Eq.(34) at the nodes on dRUdD and Eq.(35) at the nodes on d D , results in a system of 3 M nonlinear equations with 5 M unknowns, say A(:)% = b, where 14 contains the unspecified values of $1 = $2 on d D , d $ l / d n on dR and d & ? / d n *on d D , and g = (xj,yj) for j = 1,M are the two-dimensional Cartesian coordinates of the boundary element nodes on dD. For a given initial guess of g this system of equations becomes linear and it can be solved t o determine the (calculated) current flux data d 4 l / d n on do. We can then

132

minimize

However, even by minimizing the functional ( 3 6 ) ,the above system of equations is still underdetermined having 3M equations with 4M unknowns. Additional information is therefore necessary in order to account for the illconditioned nature of the discretized inverse problem. Such constraints (additional information) may include: (i) a: E R , such that the unknown object D is always contained in 52. (ii) The inclusion in (36) of penalty regularizing terms such as X l l l ~ 1 1 ~ , X211~'11~ or X311:"112, where XI, Xa, A3 > 0 are regularization parameters which may allow for continuous, C1 or C2-boundaries d D , which also stabilize the numerical solution. (iii) The boundary d D is the union of two disjoint graphs of functions, say y1 = y'(z), y2 = y2(z), z E [u,b],such that the number of unknowns is reduced by M , with only the components yj for j = 1, M needed t o be recovered. All these situations will be numerically investigated in a future work following the lines of Lesnicll for an isotropic conductivity. Moreover, the recent reconstruction methods of Ikehata13>14can also be considered numerically using the boundary element method proposed. So far, preliminary numerical studies showed that elliptical inclusions can be uniquely retrieved from a single boundary measurement , but the theoretical proof still remains a conjecture. 5

Conclusions

In this paper the inverse conductivity problem which requires the determination of the location, size and/or non-dimensional anisotropic conductivity, K , of a circular inclusion D contained in a domain R from measured electric voltage, $, and electric current flux, on the boundary a R has been investigated. The proofs of the theorems quoted in Ikehatal as exercises for the reader have been provided and various examples have been discussed. Furthermore, a boundary integral representation has been developed and a boundary integral method combined with a constrained minimization procedure have been setup for a future numerical implementation.

2,

133

References

1. M. Ikehata, Size estimation of inclusion, J. Inv. Ill-Posed Problems 6 , 127-140 (1998). 2. V. Isakov, Commun. Pure Appl. Math. 41, 865 (1988). 3. M. Ikehata et al, Appl. Anal. 72, 17 (1999). 4. O.A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics (Springer-Verlag, Berlin, 1985). 5. G. Alessandrini and R. Magnanini, Elliptic Equations in Divergence Form, Geometric Critical Points of Solutions, and Stekloff Eigenfunctions, SIAM J. Math. Anal. 25, 1259-1268(1994). 6. Hyeonbae Kang, Jin Keun Seo and Dongwoo Sheen, The Inverse Conductivity Problem with One Measurement: Stability and Estimation of Size, SIAM J. Math. Anal. 28, 1389-1405(1997). 7. G. Alessandrini and V. Isakov, Rend. Istit. Mat. Univ. Trieste 28, 351 (1996). 8. H. Kang and J.K. Seo, Identification of domains with near-extreme conductivity: global stability and error estimates, Inverse Problems 15, 851867( 1999). 9. H. Ki and D. Sheen, Numerical inversion of discontinuous conductivities, Inverse Problems 16, 33-47(2000). 10. R. Duraiswami et al, Eng. Anal. Boundary Elem. 22, 13 (1998). 11. D. Lesnic, A numerical investigation of the inverse potential conductivity problem in a circular inclusion, Inverse Problems in Engineering 9( l),l17,(2001). 12. Y.P. Chang et al, Int. J. Heat Mass Transfer 16, 1905 (1973). 13. M.Ikehata, Reconstruction of inclusion from boundary measurements, J. Inv. Ill-Posed Problems 10, 37-66(2002). 14. M.Ikehata, On reconstruction in the inverse conductivity problem with one measurement , Inverse Problems 16, 785-793(2000).

ON STABILITY ESTIMATE FOR A BACKWARD HEAT TRANSFER PROBLEM JIJUN LIU Department of Mathematics, Nanjing Normal University Department of Applied Mathematics, Southeast University Nanjing, 210096, P .R. China E-mail: [email protected] The author considers an inverse problem for 1-D heat transfer problem with variable coefficients and the Robin boundary. Our aim is to determine the initial temperature distribution from the noisy data measured at some final time T > 0. We establish a stability estimate and the uniqueness for this inverse problem, under a-prior knowledge for the solution. Furthermore, a regularization scheme, as well as the convergence rate, is proposed based on this stability result.

1

Introduction

Let R = (0, 1),QT = R x (0, TI. Consider the 1-d parabolic system &u = &(u(x, t)&u) - q ( X , t)u, (2,t ) E -azu(o, t ) hu(0,t ) = 0, t > 0 , &u(l,t) H u ( 1 , t ) = 0 , t > 0, 4 2 , O ) = f(z), 2 E [O, 11,

+ +

QT,

(1)

where h, H 2 0 are two known constants. Let p E ( 0 , l ) be a constant. For the coefficients a ( % t, ) ,q(x,t ) , we assume that ( H l ) . u(z,t)2 a0 > 0,u(2,t),q(Z,t),uz(z,t) E Cf17g(aT),then from the standard theory of linear parabolic equation’, we know there exists unique solution u ( z , t )E C2+fli1+g(QT) for f(x) E D ( f ) with

D ( f ) = {q5(z) : q 5 ( ~ ) E C2+’(Q,-q5‘(0)

+ hq5(O) = 0, q5‘(1)+ Hq5(1) = 0).

Moreover, there exists a constant C1 = C1( a ,q , h, H , Q T ) such that

holds for all f(x) E D ( f ) . Here we apply the standard Sobolev space CSk+flyk+g(aT) for k = 0 , l and C2+fl(O).That is, C2k+fl9k+$(QT)

= {u : 21 E C 2 ” k ( Q T ) , ~ f l , ~ ( L ) T D : 0 which implies that

5 Olnp(t1) + (1 - 0) Inp(t2)

138

for any 0 5 8 I 1 and t l , t 2 E [0,TI. Now we take 8 = t / T and tl = T ,t 2 = 0 in this estimate, then the above inequality generates our result immediately. Let the admissible set for the initial function f(x) be

for some known constant m > 0. Then the following result is obvious from Lemma 1 due to the fact f E prn implies I l f l l L 2 ( n ) I m.

Lemma 2 For f (x)E prn, the solution u(x,t ) t o (1) satisfies

for all 0 5 t 5 T , where we set

E

= 11g11 lm.

The above estimate is true at t = 0 but nonsense. Our main result, the relation between u(x,0) and g(x) is given by

Theorem 1 For the solution u ( x , t ) to ( 1 ) corresponding t o f(z) E pm, it follows

for

E

> 0 small enough.

Proof: Firstly, expanding llu(t)112at t = 0 says

for all t E [0,TI from Lemma 2. On the other hand, it follows from ( 2 ) that

Therefore the above estimate leads t o

+ 2mM&lTt < rn2c2t/T + 2 m M t

llu(o)112 I rn2EZtlT

139

for

E

> 0 small enough.

By elementary computation, we get

1

min (m2c2tlT+- 2mMt = -mMT

tE[O,Tl

l-ln-MT

mlne

In€

which completes our proof due to M = Clm.

Remark 2 Under a restrictive condition u ( x ,0 ) = f (x)E ,urn,this theorem asserts that u(x,O) depends on the final value u ( x , T ) = g(x) in a weak topology, that is, we measure both u(x,0 ) and u ( x ,T ) by L2(R)-norm,rather than b y C(2+fl)(n)-norm. This is due to the ill-posedness of our inverse problem. We do not know weather the L2-norm can be improved an our stability result. Now the conditional stability for our inverse problem (1) can be obtained from the above lemmas immediately: Theorem 2 Let ui(x,t ) solve (1) with f = fi E pm for i = 1,2. Then IK1.

where

I

- u2)(0)11~-

= Ui(x,T),Eo=

1

l - l n e 4m2~ In €0

9

(13)

1191 - 9211.

From this stability estimate, the uniqueness of the backward heat transfer problem up to the initial time t = 0 can also be obtained.

Theorem 3 Let u i ( x , t ) solve (1) with f(x) = fi(x) E prn for i = 1 , 2 . Then fi(x) = f2(x) in c2+P(R) i f g 1 ( x )= g2(x) in L ~ ( R ) . Proof: It is obvious from (13) that

for g1(x) = g2(x) in L 2 ( R ) , which means fl(x) = f2(x) in C(n) due to f l ( x ) - f2(x) E c2+P(SZ) c ~ ( 0 )SO . we get fi(x) = f2(x) in C2+P(n) for fl

7

f2

E c 2 + p (Q.

Remark 3 If the coeficients in the heat equation does not depend on time variable, then the uniqueness of backward heat transfer equation is obvious under the assumption that the solution exists, since the solution can be expressed in terms of the eigenfunctions of forward heat problem4. However, in our problem, this representation is general impossible due to the time dependance of coefficients. So the uniqueness is not obvious. In this sense, our

140

stability estimate is very important both in the uniqueness and in the regularization scheme for this inverse problem. The other possible way t o get the uniqueness for the backward parabolic equation with the Robin boundary condition in QT can be obtained by a classical w a g . However, the uniqueness for u(x,O) is not obtained there.

3

Regularization and Convergence Analysis

In this section, we will apply our stability result to establish a regularization scheme so that we can determine the approximation of f o ( x ) from the noisy data ga(x). Moreover, we also give the estimate of convergence rate. Assume that the exact final temperature g o ( x ) = u o ( x , T ) is generated from some initial temperature f o ( x ) = uo(x,O) E D ( f )from system (1). Now if we get the measured data g a ( z ) of g o ( x ) with the error level 6 > 0 in the sense of ( 5 ) , we want to find the initial temperature distribution fo(z) approximately. Suppose that we have a-prior knowledge of the exact solution f o ( x ) , say f o ( x ) E pm. This means we know the bound of luo(x,t)lg, (2+P)71+g from the estimate (2). Furthermore, define a functional

over D(f).

+

Theorem 4 For any Ci > m2 1, there exists a n approximate minimizer f s ( x ) for functional F & ( f ) over D(f) which satisfies

I c;s2,

@Z(fd

(15)

I W f a - KfoIlLz(n) I (CO + 116.

(16)

Proof: It is obvious that

F66,(fo) = IlKfo - gallLz(n) + s2 (Ifol:+/3)) 2

2

2

5 1190 - gsllLZ(n) + m2h2 5 d2 + m2h2= (m2+ 1)b2, (17) which implies {f : F $ ( f ) I Cis2} # 8 due to Ci > m2 + 1. Hence (15) is proven. From this inequality we also know that fa E D(f) satisfies If6lE (2+fl)

I co,

(18)

141

IlKf6 - gsllLZ(n) I COS.

(19)

Therefore we get

11m -K

~ ~ II I I ~ ~I -(g6iiLz(n) ~~ ) + iig6 - K ~ ~ I I I(co ~ +~1)s. ( ~ )

So we get (16). The proof is complete. Now u6(x,t), the approximate value of uo(z,t) can be construct from f6(X)

by solving a forward heat problem. That is,

Theorem 5 For fa(z) E D ( f ) generated in the above theorem, we solve the forward heat conduction problem ( 1 ) with f ( x ) = fa(.) to get the approximate temperature u6(x,t ). For such a approximate solution, it holds

- uo)(t)ll,z(Q)I 2(m + 2I2btIT for all 0 < t < T , while at t = 0 , it holds that II(W

for 6

>0

(20)

small enough.

Proof: The proof can be completed from Theorem 4and Theorem 2, by taking u1 = u6 and u2 = uo respectively. Firstly, we fix CO= m+ 1for certainty. Then (18) says fa E p(m+l)which means f a - f o E pz(m+l)due t o f o E pm C p(m+l).Now (9) in Lemma 2tells us

II(% - uo)(t)llI (2(m + l))l-t/T

llKf6 - K f o l y .

for 0 < t < T . Now inserting (16) into this estimate leads t o (20). For (21), it is obvious from Theorem 2that

(22)

142

which complete the proof of (21) immediately.

Remark 4 The only information in our method is the up bound of the exact solution u o ( x , t ) at t = 0. The constant m is not dificult t o get in many cases. Further, our estimate gives the error bound b y m and S explicitly. From the convergence rate, we know that u ~ ( x , t converges ) to uO(x,t)fast near t = T and slowly near t = 0. This is reasonable f r o m the physics background. Especially, our estimate o n - uo)(O)II is a little weaker than from (H), due to the fact 1 - In + +m.

&

2

Acknowledgments The author would like to give his thanks to Prof. J.Cheng for the useful discussions on this paper. This work is also partly supported by the Science Foundation at Southeast University (No.9207011148).

References 1. J.Cheng and G.Nakamura, Stability for the inverse potential problem b y finite measurement on the boundary, to appear in Inverse Problems. 2. J.Cheng and M.Yamamoto, The global uniqueness for determining two convection coeficients from Dirichlet to Neumann map in two dimensions, Inverse Problems 16(3), L25-L30 (2000). 3. A. Friedman, Partial Differential Equations of Parabolic Type (PrenticeHall, Inc., 1964). 4. V.Isakov, Inverse Problems for Partial Differential Equations (SpringerVerlag, New York, 1998). 5. J.J.Liu, Determination of temperature field from backward heat transfer problem, Communications of Korea Mathematical Socity 16(3), 371384(2001). 6. L.E.Payne, Improperly Posed Problems in Partial Differential Equations (Regional Conference Series in Applied mathematics, SIAM, Philadelphia, 1975). 7. T.I.Seidman, Optimal filtering for the backward heat equation, SIAM, J . Numer. Anal. 33(1), 162-170(1996). 8. Qixiao Ye, Zhengyuan Li, A n Introduction to Reaction-Diffusion Equations (Science Press, Beijing, 1994).

AN EXISTENCE FOR AN INVERSE PROBLEM FROM COMBUSTION THEORY AND ITS NUMERICAL SIMULATION YICHEN MA, &I CHEN AND GENJUN YING Science College,Xi’an Jiaotong Univ.,Xi’an, 710049 E-mail: [email protected]. cn GONGSHENG LI Department of math. and phys.,Zibo Univ.,Zibo city,Shandong,255000 E-main: [email protected]. cn In this paper we are concerned with a quasilinear parabolic equation with homogeneous Cauchy and non-homogeneous Neumann conditions arising from combustion theory. By using the Schauder fixed point theorem and Green function of the second homogeneous boundary value problem, we give a local existence result to the solution of an inverse problem defined on a semi-infinite space. Numerical simulation results show that the proposed numerical algorithm is efficient and applicable.

1

Introduction

Inverse problem for a parabolic equation is an important research field of inverse problem. In particular the inverse problems concerning nonlinear parabolic equations are challenging. For example: the determination of the diffusion parameter. Since the ~ O ’ S , many scientists are interested in determining the nonlinear right-hand term of the quasilinear parabolic equation. The list of researchers includes J.R. Cannon1i2, P.C. Ducheutrau’ etc. In the paper3, S. Gatti gave the local existence proof on the solution ( u , a ) to the following inverse problem, where u is the thermal profile and a:, = ( - 0 9 , O ) x (0,T).

The physical model describes a semi-infinite one-dimensional space of homogeneous solid propellent burning in a vessel at an uniform pressure. We assume that the propellent is adiabatic except at the burning surface. Two 143

144

external sources act on the propellent: p(t) is the part deposited at the surface of an external radiant flux originating from a continuous wave source concentrated at the burning surface, while f(z,t)is the remaining part of the flux distributed volumetrically along the propellent. The function R(4)is the burning rate described by the pyrolysis law(Arrhenius law, see DeLuce4y5 for details). S. Gatti considered that the data of the inverse problem are the surface temperature at z = 0. By means of the Schauder fixed-point theorem, he proved that, for a sufficiently small T , there exists at least one solution to the inverse problem (1). A similar problem was studied by Lorenzi and Paparoni in A. Lorenzi for a semi-linear parabolic equation on a bounded domain. In this paper, we assume that (z,t ) E OT = (0,GO) x (0,T ) . The additional measured data are 637

u(xolt ) = q t )

(2)

where (xo,t ) E Q ~ , x 0> 0 is a fixed point. We then obtain the following equation which will be discussed later in this paper:

{

atu(z,t ) - azu(z,t ) f R(u(0,t))&u(z,t ) = f(u(z, t ) ) ,(2, t ) E OT z 20 ~ , u ( O , t )= F ( t , u ( O , t ) ) ,

u(z,O) = 0,

(3)

05 t 5 T

It was firstly proposed by J.R. Cannon that the finding of f ( u ) in the equation (3) with condition (2) is an inverse problem for a quasilinear parabolic equation with the special boundary condition. For a general third boundary condition, non-local boundary condition and the complex boundary condition1, the inverse problems become very difficult to be solved. In this paper, we prove an existence result based on the condition that the R(u(0,t ) ) is sufficient small. We also give the numerical algorithm and examples. 2

Assumptions

According to the theory of the parabolic e q ~ a t i o n ' ~when ~ , source term f(z, t), initial data and boundary data satisfy some proper conditions, the direct problem (3) is well-posed, i.e. there exists a unique solution. Let's introduce the spaces and norms.

So [ u ( . , t ) (= l sup,,o Iu,(z,t)(,t E [O,T].The admissible set o f f is

y = {f

E

f(v)l 5 Llu. - 4) Define l f l a = SUP,,,~D If(.) - f(.)l Iu - 7 J y

C ( D ) ,llfllc 5 El f ( 0 ) = 01 If(u)

where L , E are positive constants.

-

'

145

To obtain the expression of the solution of equation (3), the Green function” on the parabolic equation with the second homogeneous boundary value condition is

+ exp

(4)

146

R(0) = 0; IRpl, lRtl 5 C R

(11)

Suppose the admissible set of (f,p ) is scd =

where E

3

{(flp)

s, II(fip)IIS 5

(12)

> 0 is a constant.

Lemmas and the Estimations

Without loss of generality, suppose (f,p ) E S c d , 0 < t 2 < tl < T , from the properties of Green function (4) and the assumption of ( 9 ) , ( 1 0 )and ( l l ) ,it is not difficult to prove the following lemmas:

Lemma 2 (K. Yosida13) For 0 < a _< l , C o ~ a ( Dis) compactly imbedded in C ( D ) , and T > O,C1[0,T ) is compactly imbedded in C[O,T ) .

147

Lemma 4 For fixed z E (0,oo), then IUZ(5,t l ) - UZ(? t 2 ) l

5 cult1 - t211/2

where c, is a bounded positive constant for Vt1,t 2 E (0, T). Remark 1 W h e n /lRllm is suficient small, d1,dz is positive and tend to 1, as T + 0 , so C1(T),and b ( T ) tend to 0, when T -+ 0.

4

Existence of the Inverse Problem

Theorem 1 Let e ( t ) and F ( t , p ( t ) ) satisfy (1)

e(o) = e’(o) = O ; ~ , ( X ~ ,=~ o,t ) E [o,T];

(2) O ( t ) E C1~a([O,TI);

(3) F ( t ,p ( t ) ) satisfies (9).

Then there exists T * , such that T maps scd =

For given

(fi,p1) E S c d ,

fn+l

Scd

to S c d , when t

E

[O,T*], where

{(f,P), II(f,P)llS 5

define the series = Tl[fn,pnl;

pn+1 = T 2 [ f n ,Pnl

(13)

According to theorem 1, we have { ( f n , p L , ) }E S c d . If 0 < a < 1/2, from lemma 2 there is subsequence { ( f n , p n ) } strongly converging in C o @ ( D )x C’[O,TI.

148

Since formula (7)

It follows from (5)

Now we obtain a convergence subsequence and its limit f, u), It is necessary to prove the limit is the solution of the equation (3). Thus

A

u ( 2 , t )= lim u n ( x , t ) , n-oo

A A

f (u)= n-ioo lim fn(un)

We will describe the proof of the theorem 1 and theorem 2 in detail in next section. 5

The Proof of the Theorem 1 and Theorem 2

In this section,we give some lemmas and their proofs.At first,we give some basic properties of the Green function K ( z , y ; t ) ,which will be used in the following proofs.

149

Property 3

where

Property 4 There exists a constant c(x0) only depending o n xo that

Lemma 6 Under the conditions of the theorem 1, if (lo), and IIRII,is sufficient small, then

where d, = (1 - T1/211Rlloocl(0))-1> 0.

fn

> 0 , such

satisfies the property

150

Acording to assumption (9) and property 2, we have and

Thus A

To 5 CFIIPn- P

1100.

A

Because of fn E Y ,f~ Y , A

TIL c ~ ( O ) ( l l f n -f

lloo

+ Lllun-

A

u l(m)T1/2

Similarly

I I So so K Z ( z 7 y ; -t ~)[R(P)(uE-hz)+ ( R ( p n ) - R(P))$]dyd.r( A A I llRllooll4Ilooc1(0)t1/2+ ~ 1 ~ ~ ~ l l ~ ~ l l o I.Lo 110011.LLElloot1/2 llPnA

t o o

T2

A

If 1 - IIRIlco~1(0)T1/2 > 0 , the proof is completed. Remark 2 If I IRI is suficient small, d , will be positive constant depending o n T , and as T tends t o 0, d , tends t o 1. Lemma 7 Under the conditions of lemma 6, un tends t o u, as n tends A

Proof: Under the formula (5) and the expression of u , we have A

t o o

A

A

I u --UnI 5 I So So K(x:iY ; t - 7)[R(P)u, -R(pn)uE]d&( t c o A A +I so so K(z7 Y ; t - .)[f ( u )- fn(un)ldYd.rl A +I s,” K ( z ,0; t - 7 ) [ F ( 7P, ) - F ( r ,P.n)ldTl = T3

+ T4 + T5

00.

151

According the conditions of Fl R, f and the properties of Green function, we get

so.fo

t o o

T3 =

A

5 II uz

A

A

K ( x ,Y;t - T ) { [R(P)- R(pn)]21, +R(p,)[& -uF]}dyd~ A

IloollRPllooll

I-L -CLnIIooco(O)t

+ II~lloolluF-Au z Ilooco(o)t

7

then T3

where $ p , t )

i r(llpn-

A

=

II 21,

A

p

IloollRPllooco(~)t.p, so y ( p , t ) 2cF

7-5

Similarly, for

T4, we

A

llm) + IIRllooll4-

I -+IllnJ;;

uz +

lloot

1

0,as t

4

0.

A

CL lloot1'2.

have A

T4

A

I 4bn- 1' 1 lloot + Ilfn- f l l o o t . A

A

Following form the lemma 6 and limn-+oofn =f,limn-oopn =p, when T is small enough, we have lim un(x,t ) = u ( x lt)l ( x ,t ) E f2T

n-oo

The proof is completed.

Lemma 8 Under the conditions of lemma 6, if x is fixed, then

I&-

A

uxxloo 4 0, n

--f

~0

152

Similarly

Following lemma 6 and lemma 7, when n , T6 0. The proof is completed. The proof of theorem 1 Following from (7) and lemma 1 (1), we obtain

Substituting the estimations of ~ ~ u zl[uzzllm ~ ~ m into , (18), we get -

Ilfllm 5 where

+CfV)

(19)

153

where I ( t ) is defined as in lemma 2,and

so

154

155

because of

and

According to lemma 2

From property 3

Thus

156

+CF(T + IlPllco)T'/2 + II~,IlcoII~~llcollPllcoco(O)T + IlfllcoT So, we have IIT[f,pL]IIs I t $ ( E ) &E) llO)ICl,-. For given E > 2 ~ ~ O ~ ~wec can ~ , achoice l a proper T * ,such that , z(0,")

for x E (011)

(3)

and the boundary conditions

d2Y y ( t , O ) = y(t,Z) = 0, -(t,O) 8x2

d2Y = -(t,I) 8x2

=0

(4)

162

for t E (0,T). We introduce the Hilbert space V = H2(0,Z)n HJ(0,Z) with the inner product (w,z) = s," dz. Next we define the operators A , B:V + V * by ( A w , z ) = dP( w , z ) and ( B w , z ) = y ( w , z ) . Then the equation (2) with the law (1) and the specified boundary and initial conditions (3) and (4) can be written in the form: find y : (0,T)-+ V such that

3

{

y"(t)

1 + Ay'(t) + By(t)- -f(t) P

1. = - g ( t ) a.e. t E ( 0 , T )

P

-f(t) E U * ( a j ( U y ( t ) ) a.e. ) t E (0,T) Y ( 0 ) = Y o , Y'(0) = y1.

(5)

Here the operator U :L 2 ( R )+ L2(R') is defined by Uv = v l n ~where , R = (0,Z) and R' = (Z1,Zz). Its adjoint operator U*:L2(R') -+ L 2 ( R )is given by

R' (u*v)(x)= 0 otherwise. Therefore the multivalued relation in (5) is equivalent to the following two conditions -f(t,x) E a j ( y ( t , z ) ) for ( 4 2 ) E (0,T) x R' for t E (O,T),z $ 0'. -f(t,z) = 0

{

~ ( z )if z E

{

The problem analogous to (5) can be considered in the case of Kirchhoff plates (see Panagiotopoulos and Pop 1 6 ) . For other examples appearing in the modelling of linear visco-elastic materials (cf. Vol. 1, Chapter 3 of Dautray and Lions '), see Ochal 12. Let Y be a reflexive Banach space and let T : Y -+ 2y* be a multivalued operator. An operator T is said to be pseudomonotone (cf. Browder and Hess 4, if it satisfies : a) for every y E Y , Ty is a nonempty, convex and weakly compact set in Y*;

b) T is U.S.C.from every finite dimensional subspace of Y into Y* endowed with the weak topology; and c) if yn + y weakly in Y , y i E Ty, and limsup (y:, yn - Y ) 5~ 0, then for each z E Y there exists y*(z) E T y such that (y*(z),y 5 liminf ( y i , y n - z ) ~ . Let L: D ( L ) c Y + Y' be a linear densely defined maximal monotone operator. An operator T is said to be L-pseudomonotone if and only if a) and b) hold and

163

c D ( L ) is such that yn -+ y weakly in Y , Lyn -+ Ly weakly in Y * ,Y; E T(yn),~ / 7 " ,-+ Y* weakly in Y* and limsup (y/7",,yJy 5 (Y*,Y ) ~ , then (YlY*) E Graph(T) and (YE,Yn)y -+ ( Y * , Y ) y .

d) if {yn}

A single-valued operator T : Y -+ Y * is said t o be pseudomonotone if for each sequence {yn} 5 Y such that it converges weakly t o yo E Y and limsup(Tyn,yn - Y O ) Y I 0, we have (TYO,YO - Y ) Y 5 liminf(Ty,,y, - Y ) Y for all y E Y . Finally, we recall (see Clarke 5 , that given a locally Lipschitz function h: E + R, where E is a Banach space, the generalized directional derivative of h at z in the direction w , denoted by hO(z;v), is defined by hO(z;w)= 1

limsup A(h(y

y+z, tJ.0 t

+ tw) - h(y)).

The generalized gradient of h a t z, denoted by

d h ( z ) ,is a subset of a dual space E* given by dh(z) = {C E E* : h o ( z ; v )2 (C,w)E, x E for all w E E } . 3 Existence Result Let (V,11 . 11) be a real reflexive, separable Banach space which is densely and compactly embedded in a Hilbert space H . The dual space of V is denoted by V * and I . I stands for the norm in H . By (-, .) we denote the duality of V and V * . Given 0 < T < +m, we introduce the following spaces V = L2(0,T;V ) , 3t = L 2 ( 0 , T ; H ) ,W = {w E V : w' E V * } , 2 = {w E V : w' E W } , where V* = L 2 ( 0 , T ; V * )X* , E 3t and the time derivative is taken in the sense of vector valued distributions. It is well known (cf. Zeidler 18) that W C C ( 0 , T ; H )and W c C ( 0 , T ; V )continuously and W c 3t compactly. Moreover, let Y be a reflexive Banach space. The multivalued second order evolution equation under consideration is the following: given yo E V , y1 E H and f E V * , find y E V such that y' E W and

{

+

+

+ N * ( d J ( t ,N y ( t ) ) )3 f ( t ) a.e. t E ( 0 , T )

y"(t) A(t,y ' ( t ) ) B y ( t ) y(O) = yo, y'(O) = y1,

(6)

where A: (0, T ) x V + V * is a nonlinear operator, B:V -+ V * and N : H + Y are bounded linear operators, d J : (0, T ) x Y -+ 2Y* is the generalized (Clarke) gradient with respect t o the second variable of a locally Lipschitz function J ( t , .): Y -+ R and N * denotes the adjoint operator of N . We remark that the initial conditions in (6) have a sense since the embeddings 2 C C(0,T ;V ) and W c C(0,T ;H ) are continuous. We say that y E V is a solution of the problem ( 6 ) with yo E V and y1 E H if y' E W and

164

there exists [ E X such that

+

+

A(t,y‘(t)) B y ( t ) + [ ( t )= f ( t ) a.e. t E (0 ,T ) Y(0) = Yo, Y’(0) = Y 1 , [ ( t ) E N * ( a J ( t , N y ( t ) ) ) a.e. t E (0,T). yl”(t)

(7)

Remark 1 The problem (6) is equivalent to the following inequality: find y E V such that y‘ E W and

{

(Y”(t)

+ A(t,Y’(t))+

-

f ( t ) ,4 + J 0 @ ,N y ( t ) ;NU) 2 0

for a.e. t and all w E V

Y(0) = Yo, Y’(0) = Y 1 ,

where J o ( t ,z ; w)is the generalized directional derivative of J at a point z E Y in the direction w E Y . This justifies the name hemivariational inequality given to problem (6). We make the following assumptions:

H ( A ) : A: (0,T)x V

+ V * is an operator such that

(i) t ++ A(t,w) is measurable on ( 0 ,T ) ; (ii) w H A(t,w) is pseudomonotone for each t ; (iii) IIA(t,w)lIv* 5 a l ( t )

+

blllwll a.e t E (O,T),for all w E V, al E L 2 ( 0 , T ) , 2 0 , bl > 0; (iv) ( A ( t ,w),w) 2 /3111w112 a.e. t , for all w E V with PI > 0.

H ( B ) : B: V

-+ V * is a linear, bounded, positive and symmetric operator.

+ Y is a linear and bounded operator. H ( J ) : J : ( 0 , T ) x Y + B, J = J ( t , z ) is measurable in t E (O,T),locally L i p s c h i t z in z E Y and for C E a J ( t , z ) we have IICIIy. _< ~ ( +1 IlzIIy) for H ( N ) : N :H

every z E Y , t E ( 0 ,T ) with E (Ho) : YO E ( H I ):

> 0.

V , y1 E H and f E V * .

81 > 4P2TF llN1I2,where /3 is an embedding constant of V into H

IlNll = IINIIL(H,Y,.

and

165

Lemma 1 If hypotheses H ( A ) , H ( B ) , H ( N ) , H ( J ) and ( H o ) hold and y is a solution to (6), then there exists a constant C > 0 such that IlYllC(0,T;V)

+ IlY‘llW L C ( 1 + IlYOll + lY11 + Ilfllv*).

Lemma 2 Assume hypotheses H ( N ) and H ( J ) . Then the multivalued map R: (0, T ) x H + 2 H defined by R ( t ,v) = N * ( d J ( t ,Nw))has nonempty, convex and weakly compact values in H , R ( t , . ) is from H into Hweak and there is a constant C > 0 such that IR(t,v)l 5 C(l+ 1.1) for all v E H . Theorem 1 Under hypotheses H ( A ) , H ( B ) , W ( N ) ,H ( J ) , ( H o ) and ( H I ) , the problem (6) admits at least one solution. Proof: We present the main idea of it. First we reduce the order of the problem (6). We consider the operator K : V + C ( 0 , T ;V )defined by K v ( t ) = w ( s ) ds yo for w E V . The operator K is bounded and continuous from V into C(0,T; V ) . Using K we can write the problem (6) as follows: find z E W such that

+

{

+

z’(t) A(t,z ( t ) )+ B ( K z ( t ) )+ R ( t ,K z ( t ) )3 f ( t )a.e. t E (0, T ) 4 0 ) = y1.

(8)

Now we can see that z E W solves (8) if and only if y = K z solves (6). Therefore, it is enough to prove the existence of solutions to (8). We consider two cases: first we study the problem (8) with regular initial condition y1 E V and then we deal with a general case y1 E H . In the first case we define the following operators d1:V + V * , B1: V + V* and 7 2 1 : V + 2’’ by ( d 1 ) ( .= ) A(.,w(-) y l ) , (&)(.) = B(K(w y l ) ( . ) ) and Rlw = { w E 3t : w ( t ) E R(t,K(w y l ) ( t ) ) a.e. t for all w E V , respectively. Here w y1 is understood as follows (v y l ) ( . ) = w(.) y 1 . Exploiting the above operators the problem (8) is formulated in the following way

+ +

+

+

+

c

+

+

z’ d l z + B I Z R l z 3 f z ( 0 ) = 0.

+

a.e. t E (0, T )

(9)

Note that z E W solves (8) if and only if z - y1 E W solves (9). Next, introducing the operators L: D ( L ) c V -+ V* and T :V + 2v* given by L z = z’ with D ( L ) = { z E W : z ( 0 ) = 0) and T z = d l z + & z + R l z , respectively, the problem (9) takes the form: find z E D ( L ) such that Lz T z 3 f . In order t o establish the existence of a solution t o (9), we use a surjectivity result of Papageorgiou, Papalini and Renzacci 17. To this end, exploiting a result of Berkovits and Mustonen and the Convergence Theorem of Aubin

+

166

and Cellina ' ,we are able to prove that T is a bounded, coercive and Lpseudomonotone operator (cf. the definition in Section 2).

Example 2 Let R C IRN be a bounded domain with a Lipschitz boundary and let Y = L2(R;R N ) . We consider a function j : ( 0 , T )x R x RN -+R such that

H ( j ) : j ( . , . , v ) : ( O , T ) x R + Rismeasurableforallv E I R N , j ( t , . , O ) E L1(R), j ( t ,2,.): RN + R is locally Lipschitz for all ( t ,x) E ( 0 ,T ) x R and for all

5 E a,,j(t, x,v) we have I I ~ J w N

5 c (1 + I I w ~ ~ R N ) with c > 0.

s,

We define J : (0,T ) x Y + IR by J ( t ,v) = j ( t ,x,v(x)) dx. It can be verified that if the integrand j satisfies H ( j ) , then the functional J satisfies H ( J ) . Furthermore, we can easily see that if N = 1 and B E LEc(R) is such that Ip(s)l 5 c(l+lsl)fors E R, thenj(t,x,v) = $',L?(s)dssatisfiesthehypothesis H(j)4

4.1

Optimal Control for Hemivariational Inequalities Bolza Type Optimal Control Problem

We consider a system described by the following controlled second order evolution inclusion:

+

+

+

y " ( t ) A ( t ,y ' ( t ) ) By(t) N * ( d J ( t ,N y ( t ) ) )3 f ( t ) y(O) = Y o , y'(O) = 91,

+ C ( t ) u ( t )a.e. t

(10) where y = y ( u ) is the solution corresponding to a control variable u E U = L 2 ( 0 , T ; X ) ,X being the space of controls, C represents a controller and A , B , N , J , f , yo and y1 are as in the previous section. We deal the following Bolza type optimal control problem ( C P ) : @ ( y , u ) + inf, where y E S(u) and E U ( t ) a.e. t E (O,T),u(.)is measurable

{ u(t)

and the cost functional is given by

We admit the following assumptions:

H ( C ) : C E L"(0, T ;L ( X ,H ) ) and X is a separable reflexive Banach space.

167

H ( @ ): 1: H x H + R is weakly lower semicontinuous; F : [0,TI x H x H x X R U { +oo} is a measurable function such that

+

(i) F ( t , ., ., .) is sequentially lower semicontinuous; (ii) F ( t , y, z , .) is convex; (iii) there exist M

> 0 and + E L1(O,T)such that F ( t ,y, z , u ) 2 +(t)-

+ 14 + Il.llx). H ( U ) : U : [0,T ]-+ 2x \ {0} is a multifunction M(lYl

is a closed convex subset of X and t

Ly .

such that for all t E [0,TI, U ( t )

+ sup{ 1 ) u ) :) ~u E U ( t ) }belongs t o

Theorem 2 I f t h e hypotheses H ( A ) , H ( B ) , H ( N ) , H ( J ) , ( H o ) ,( H I ) ,H ( C ) , H ( @ )and H ( U ) hold, then the problem ( C P ) admits an optimal solution.

For other control problems for systems modeled by (lo), a time optimal control problem and a maximum stay control problem, we refer t o Ochal l2 and Migorski *. 4.2

The Identification Problem

We consider the parameter estimation problem for the hemivariational inequality model (6). We state this problem in terms of finding parameters which give the best fit of the parameter dependent solutions of hemivariational inequality t o the observation data for response of the system t o excitations. Let the collection of unknown parameters be denoted by p and we assume that it belongs to some admissible parameter set P. Given p E P we denote by S ( p ) the solution set of

The formulation of the inverse problem is as follows: given a cost functional F = F ( p ,y), F : P x 2 -+ find p* E P and y* E S ( p * ) such that

F(P*,Y*) = inf{F(p,y)

:P E

p , Y E S(P)}.

(12)

We admit the following hypotheses: h

H ( P ) : P is a compact subset of a metric spaces of parameters P ,

168

H ( A ) l : for any p E P , A ( p ) E C(V,V*),(A(p)v,w)1 c111w112 for all w E V with c1 > 0 independent of p and p , -+ p in ? implies A(p,) -+ A ( p ) in Cc(V7V * ) ; H ( B ) 1 : for any p E P , B ( p ) E C(V,V " ) ,B ( p ) is symmetric and positive and p , -+ p in ? implies ~ ( p , -+ ) ~ ( pin) C(V, v*); H ( J ) 1 : for any p E P , J ( p ) :(0,T) x Y -+ R is measurable in t E ( 0 , T ) and locally Lipschitz function in w E Y such that (i)

11511~*I c2 (1+ IlwIIy) for 5 E a J ( p > ( t , v ) 21, E H with

(ii) if p , -+ p in

c2 2 0, ?, then lim sup Gr a J ( p , ) ( t , .) c Gr d J ( p ) ( t ,.) in Y

x

n-+m

Yweak

topology, for all t E (O,T),

where G r d J ( p ) ( t , . ) = { ( z , w ) E Y x the graph of a J ( p ) ( t ,.).

Y * :

w E a J ( p ) ( t , z ) } stands for

Theorem 3 If hypotheses H ( P ) , H(A)1, H ( B ) 1 , H ( J ) 1 and ( H I ) hold, ( y 0 , y l ) E V x H , f E V" and F as lower semicontinuous in P X 2 w e a k topology, then the problem (12) admits a solution. We remark that the hypothesis H ( J ) l ( i i ) holds, for example, if J(p,)(t, .): Y -+ R, n 11, are locally Lipschitz, equi-lower semidifferentiable, locally equi-bounded and J(p,)(t, .) 4 J ( p ) ( t ,.) for all t E (0, T ) (see Theorem 1 of Zolezzi ''). In the examples, we may consider the problem of estimating of parameters by fitting data w obtained from displacement, velocity or acceleration measurements at various locations in a body R C RN . This leads to functional F ( p ,y ) = G ( y ) I ( p ) , where I:P -+ R is a lower semicontinuous on P and G: 2 -+ R is of the form

+

G(Y)=

2

( J l Y ( t i ; P )-W2'll:,

+ J J Y l ( t i ; P-) wsll;)

i= 1

or

( I y ( x ,t ) - w3I2

G(y)=

+ Iy'(x, t ) - w4I2)d x d t

1

( E l = rl x (O,T), where 0 < tl < t 2 < w4 are fixed targets.

c dR, m ( r l ) > 0) subject t o y = y ( - ; p )satisfying ( l l ) , . . . t , 5 T are points of measurements and w:, w l , w3,

169

Acknowledgments The research was supported in part by the State Committee for Scientific Research of the Republic of Poland (KBN) under Grants No. 2 P03A 004 19 and 7 T07A 047 18.

References 1. J. P. Aubin and A. Cellina, Differential Inclusions. Set-I ilued Maps and Viability Theory (Springer, Berlin, New York, Tokyo, 1984). 2. H. T. Banks, R. C. Smith and Y. Wang, Smart Material Structures: Modeling, Estimation and Control (Wiley, Chichester, Masson, Paris, 1996). 3. J. Berkovits, V. Mustonen, Monotonicity Methods for Nonlinear Evolution Equations, Nonlinear Anal. 27, 1397-1405 (1996). 4. F. E. Browder and P. Hess, Nonlinear mappings of monotone type in Banach spaces, J. Funct. Anal. 11, 251-294 (1972). 5. F. H. Clarke, Optimization and Nonsmooth Analysis (Wiley Interscience; New York, 1983). 6. R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol.1, Physical Origins and Classical Methods (Springer-Verlag, Berlin, 1992). 7. S. Migbrski, Existence and convergence results for evolution hemivariational inequalities, Topological Methods Nonlinear Anal. 16, 125-144 (2000). 8. S. Migbrski, Evolution hemivariational inequalities in infinite dimension and their control, Nonlinear Anal. 47, 101-112 (2001). 9. S. Migbrski, O n existence of solutions for parabolic hemivariational inequalities, J. Comp. Appl. Math. 129, 77-87 (2001). 10. S. Mig6rski and A. Ochal, Optimal control of parabolic hemivariational inequalities, J. Global Optim. 17, 285-300 (2000). 11. Z. Naniewicz and P. D. Panagiopopoulos, Mathematical Theory of Hemivariational Inequalities and Applications (Marcel Dekker, Inc., New York, Basel, Hong Kong, 1995). 12. A. Ochal, Optimal Control of Evolution Hemivariational Inequalities (PhD Thesis, Jagiellonian Univ., Cracow, Poland, p.63(2001)). 13. P.D. Panagiotopoulos, Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions (Birkhauser, Basel, 1985). 14. P.D. Panagiotopoulos, Coercive and semicoercive hemivariational in-

170

equalities, Nonlinear Anal. 16, 209-231 (1991). 15. P.D. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechanics and Engineering (Springer, Berlin, 1993). 16. P.D. Panagiotopoulos and G. Pop, O n a type of hyperbolic variationalhemivariational inequalities, J. Applied Anal. 5 , 95-112 (1999). 17. N.S. Papageorgiou, F. Papalini and F. Renzacci, Existence of Solutions and Periodic Solutions for Nonlinear Evolution Inclusions, t Rend. Circ. Mat. Palermo 48, 341-364 (1999). 18. E. Zeidler, Nonlinear Functional Analysis and Applications 11 A / B (Springer, New York, 1990). 19. T. Zolezzi, Convergence of Generalized Gradients, Set-Valued Anal. 2 , 381-393 (1994).

ESTIMATION OF ALL MATHEMATICAL MODEL PARAMETERS AND EXPERIMENT INFORMATIVENESS M. R. ROMANOVSKI C A D / C A E Department, POINT Ltd., 79-1-334, Schelkovskoe shosse, 107497, Moscow, Russia E-mail: [email protected], URL: http://mywebpage.netscape. com/mromanovski/IP. htm The conditions providing the highest achievable informativeness of experiment processing are examined. It is proved that a realization of a single experiment is sufficient to identify all phenomenological properties of a test object described by a superposition of mutually commuting operators. Simultaneous identification both model coefficients and boundary conditions is considered for the first time. The practical meaning of the study consists in the approach development guaranteeing the experiment against unidentifiable states and observation.

1

Introduction

Let us consider the problem of determining of the maximal information on properties of a test object during an interpretation of observation data. The main purpose is to define the maximum number of unknown parameters of an input signal, which is conveyed by a received signal. For the sake of the following terminology shortness, we will define the amount of data concerned in a sampling as an informativeness of a n observed event. This will understand as the permissible maximal volume of useful information on input signal components that is contained in a given received signal and can be unambiguously reconstructed during further observation processing. Here we will deal with the qualitative aspect, related to finding the upper bound of the informativeness, as well as with cases of the proper information degeneration. The question response will be grounded on the condition determination that breaks down the one-to-one correspondence between a direct problem solution and its coefficients. In the theory of inverse problems many authors studied the conditions to identify more than one model parameter These investigations deal with uniqueness of inverse problem solutions to substantiate a correct mapping of sought functions. In contrast to these studies, we consider the problem to extend a number of desired quantities as much as possible. A similar viewpoint is directed to practical problems to give ample opportunities of a correct identification of test object properties without numerous measurements. 19233,4.

171

172

2

Violation of One-to-one Correspondence

Let us define the permissible upper bound of the informativeness and establish the highest achievable volume of object properties that can be obtained having observation data of a single state function. Consider the following abstract model P

k=l

where a k denotes a phenomenological object’s property that accurate within the equivalence describes an input-output mapping, p is a number of model parameters, u is a state function, f is an input function, L k is an operator. Equation (1)defines in a general form a known relation between a directly observed state u,an external impact f , and object’s properties a = { a k } + G . This equation with initial-boundary conditions conveys a typical direct problem. In particular, the models of control systems as a rule are added up to Eq. (l),where ak is the equivalence combination of virgin system parameters. A lot of distributed parameter systems also give rise to Eq. (1). It is assumed that Eq. (1) has a unique and stable solution u with fixed a and f . Also, the domain of operator La

P

akLk

is supposed to be inde-

k=l

pendent of the sought quantities. The dependence of the boundary conditions on the sought quantities will be considered separately in section 3. Let us suppose that model (1) is specified as a superposition of mutually commuting operators, i.e., V i , j E [l,p] : LiLj = LjLi. In this case our main result is given in the following theorem’.

Theorem 1 To determine all properties of a test object described by Eq. (l), where ak = c m s t it is necessary and suficient t o perform a single experim e n t , in which the object state function does not satisfy the linear dependence condition P k=l

where B = { p k = c m s t 7 3 i , j E [l,p] : ,L?,,pj # 0). Theorem 1 conveys only the basic possibility for a number of unknowns to be simultaneously identified. It is also necessary to define the properties of the function f * generating the coefficient invariance. This will allow us to

173

answer the question what happens to inverse problem solutions during the violation of the one-to-one correspondence. If the operators { L k } k = i ; ; ; are mutually commuting, then acting on the two parts of Eq. (1) and adding up the terms multiplied by the coefficients we get a linear dependence condition for the image space elements of Eq. (1). Therefore, condition (2) defines the subspace, whose elements being mapped accordingly to (1) with commuting operators retain the linear dependence condition. Further, expressing any term of Eq. (1) from condition (2) we reveal the form of the non-unique subset of unknown quantities as well as the equation whose solution satisfies the linear dependence condition (2). This results in the following corollary.

Corollary 1 If the solution of Eq. (1) with mutually commuting operators satisfies condition (2), then the equation coefficients belong t o the family the input function also satisfies the linear dependence condition

f:

PkLkf

* = 07 P k E

B7

(4)

k=l

and the state u* is determined as the solution of the equation P-1 bkLkU*

=Ppf *

(5)

k=l

Hence, family (3) conveys the non-uniqueness character of the mapping from the space of object states into the space of the Eq. (1) coefficients. All its elements generate the only solution u* that satisfies Eq. (5) with reduced order in respect to the initial equation. One-to-one correspondence is violated but under certain values of f *, which are to satisfy condition (4). The violation occurs due to the linear dependence of the mathematical model terms. This dependence generates the subspace of direct problem solutions U * , whose elements correlate with the subset of the sought object properties A* determining the inverse problem solution accurate to family (3). In the above case the higher-order operator L, is singled out, so that family (3) sets the one-parameter dependence of Eq. (1) coefficients relative to a,. Expressing latter in the terms of family (3) one can prove that another form of one-parameter dependence with the same non-zero values B k E B but found with different terms of Eq.(2) is equivalent to the initial ambiguity family. If the number of the properties is p > 2, then new linearly dependent

174

terms in Eq.(5) can be selected. Hence, there is a two-parameter family and the order of Eq.(l) is reduced once more. As a result, the non-uniqueness subset of Eq.(l) contains coefficient families of one- up to ( p - 1)-parameters. Note, that condition (4) holds for arbitrary B k , when model (1) is homogeneous, f 0. Therefore, the one-parameter family from the subset A* has the form a k / a p = B k , and the coefficients of a homogeneous equation, the boundary conditions of which do not depend on their values, can be only found within the ration B k . This commonly known property of equivalence is coupled by the results quoted. As it seems, aside from a one-parameter family, homogeneous Eq.(l) leaves room for two-parameter up to ( p - 1)-parameter family of the model coefficients. Theorem 1 and Corollary 1 give the complete answer the foregoing question about the useful information contents extracted during the experimental data processing. Namely, a single experiment can provide simultaneous estimation of all phenomenological properties of the test object, if the appearance of the state u* satisfying the linear dependence condition of the initial equation terms is excluded. It is hence possible to identify the object properties starting from the observation with limited number of measurements. On the other hand, the results obtained attest that there exists a class of a direct problem solution u * , completely retained, if the free term is effected according to (4)while the equation coefficients should satisfy (3). For the reason the variance of object properties and characteristics does not change the direct problem solution at every point of its variable domain. The result obtained conveys the general functional properties of mathematical models that generate the non-unique correspondence between a direct problem solution and its coefficients. The properties’ manifestation is determined by specifying certain boundary conditions and external impacts. We will now study their determination based on the next equation alLlu

+ a2L2u = f

that is assumed to be given in a variable domain Q with boundary G1 U G2, on which the solution u satisfies the conditions = pi, i = 1 , 2 ,

(6)

6’9 =

(7) where L1,Z are given linear operators, K1,z are the corresponding linear operators, one of which, for example K1 , determines the Cauchy conditions on the boundary G I ;f and pl,2 are known functions. It is assumed that there exists a unique function u satisfying ( 6 ) and (7) while the functions f and p1,2 are smooth enough so that the values of L I Jf,K1 f I G ~ ,Kz f I G ~ , L l p z , L2p1 can be determined. In this case the following theorem holds. KiUIGi

175

Theorem 2 The one-to-one correspondence between the coeficients a1,2 and the solution of problem (6), (7) breaks down, if and only if its solution is the function u* = b-lLT1 f , f o r the existence of which it is necessary and suficient the adjustment of boundary conditions Kl(P2lG1

= K2(Pl(Gz7

(8)

the free t e r m f * must satisfy the equation

PLlf* = Laf*,P # 0

(9)

with the conditions PKlf*(G1= bL2pl K2f

*IG~

= bLi(P2

and the coeficients are given from the single family a1

+ Pa2 = b,

(12)

where b and ,B are the parameters of the ambiguity subset. The theorem proof is carried out similar Romanovskii'. From a practical viewpoint the following corollary is important.

Corollary 2 If the conditions (8) and (9) hold, then the one-to-one correspondence is provided for the homogeneous conditions Llp2 = 0 and L291 = 0 o r K l f * l G l = O a n d K z f * l ~= O~. These conditions guarantee the preservation of the one-to-one correspondence, if one designs experimental conditions. Satisfaction of any of the corollary 2 conditions ensures the absence of the unidentifiable class u* of direct problem solutions. Analysis of the conditions (9)-( 11) with terms identically zero gives the following affirmation.

The existence of the similar kind of solutions means that for every a E A problem ( 6 ) , (7) has linearly dependent terms L1,2u. In this case the subset of non-uniqueness is the entire original set of coefficients, A* 3 A, and for the

176

reason it contains an infinite number of families (12). This does not contradict theorem 2, since the infinity of families is associated with different solutions of problem (6), (7) and each of them has the unique family (12). Formulation dU

al-

at

UltZ0

d2U

= a2-

8x2

+ f,0 < x < 1, t > 0;

::

= z ( x - l ) , --

= 1,

-El

= -1 s=l

exemplifies the conditions of corollary 3. Every solution of this direct problem with f = const # 0 is defined by formula (13). So simulated field u ( x ,t ) does not provide the unique inverse problem solution for any thermal properties a1,2 and observation data. This example demonstrates, that the violation of the one-to-one correspondence between the direct problem solution and sought quantities must be kept in mind as an important problem. The results obtained convey the invariant properties of linear equations. Nonlinear equations have also the violation of the one-to-one correspondence. For example, in the class u ( x , t ) E C2y1a solution of the equation

du

at that satisfies the condition formation a: = a;

d ax

= exp

is invariant relatively trans-

+ exp

, a; = a:

+ h, where h(u)

is a displacement, the function p satisfies

+

+

+

+

and ~ ( xt ),= CO C1x C2t or r(z, t ) = CO (CI x ) / d m , CO--3 are arbitrary constants. The similar kind of the heat equation solutions are the scaling solutions ?. Thus, if we want to reconstruct several unknown quantities, then the invariant properties have to be taken into account as an important part of the uniqueness investigation. Previously, we have studied the informativeness depending on the number of experiments necessary to define all the model coefficients. We are now to see the conditions that added to a discrete set ui = ulEi on a measurement design {Ei}i=to assure the identification of the maximal number of unknown

177

quantities. This question will be analyzed within the previous mathematical framework. The following affirmation is valid. Theorem 3 Equation (1) as identifiable as to the parameters { a k } k = f i , if

both the discrete set of observation { u ~ } ~ = Gand free term f exclude the P

satisfaction of the linear dependence conditions

piLkUlzi

= 0, k =

i=l P

pi f

Izi

= 0 respectively, where

,6k

= Const, g i , j E [ l , p ]: pi,pj

G,

# 0.

i=l

The result obtained indicates that, generally, there are points at an observation domain, where no information of sought properties can be found against high measurement precision. The existence of such sensor locations is commonly known in the theory of oscillation. Theorem 2 is t o confirm that the informativeness degeneration shares a common property of mathematical models. Therefore, it is necessary t o expect the existence of such sensor locations for other kind of measurements, which do not ensure, for instance, the identification of thermal properties. To exemplify the similar situation we consider the mathematical model

UIt,O

= 210, 4,,lJ = 211, Ulx=l - 212 -

(15)

where the coefficients a1.2 = const are the sought quantities. For the case f,U O , w1,2 = const the linear dependence conditions, pointed by theorem 3, take place, if the condition exp

[-2(7)

(t2 -

t.)] =

sin x2 sin x1

9, k = 1 , 2 , ..., X # O 9

is satisfied. Then the sought coefficients a1,2 cannot be uniquely determined on a discrete sample ud = u*(xr,ti)li=1,2, if, and only if, the measurements are made at any two locations x ; , ~ such that x ; xa = l and also executed a t the same moment, tl = t 2 . So, we can give the following final answer to the question posed above. First, it is necessary and sufficient t o perform a single experiment satisfying the certain requirements to reveal a maximum volume of information on properties of a test object. In the class of the linear abstract models their state functions and corresponding observation data must meet the conditions of linear independence of the model terms. To hold these conditions in practical situations a number of simple requirements must be fulfilled.

+

178

Second, strictly defined boundary conditions and external impacts are necessary to break down the one-to-one correspondence between the state function and its coefficients. There exist models all of whose states are unidentifiable on a whole. Also, there are observation points, where no information of sought properties can be found against high measurement precision. Summarizing this part of the investigation, the following general consequence can be made. A single experiment can convey information on all the test object properties accurate within the equivalence and there are conditions to identify them uniquely.

3

Simultaneous Identification of Model Coefficients and Boundary Conditions

Let us now analyze typical experiments from a viewpoint of the maximal informativeness with absolute minimum of input data. Here we want to extend the question studied and pose the new problem of a reconstruction both model coefficients and boundary conditions grounding on a sole observation point. Consider the direct problem (14), (15). It is required to establish the existence and features of the design 9 = {zi,tj}!z==,"" with one observation point and n measurement times for which the known discrete set (sampling) ~ f = ,E ( ~z i l t j ) + ~ j i ,= 1 , j = fi allows to identify the constant unknowns a = { a 1 , 2 , ~ 1 , 2 } simultaneously. Here E denotes a measurement noise, about which we only know its upper bound, max l ~ j 5 l 6. We reduce the problem posed to the determination and analysis of the behavior of estimation errors p 1 , 2 = (Z1,z - U I , Z ) / & , Z , p 3 , 4 = ( G , 4 - ~ 3 , 4 ) / V 3 , 4 , where Z = { h l , 2 , 2 1 1 , 2 } denotes the actual values of sought quantities. Being grounded on the approach that provides the comprehensive analysis of estimation errors, one can obtain the following system

'

179

To estimate the unknowns a = { a 1 , 2 , v 1 , 2 } four measurement times( n = 4) for any sensor location €1 should be fixed. The sampling 6 j = 1 , 4 is the minimal volume discrete set of observation for the problem studied. The desired solution p i - 4 does not exist, if the observation is made at one of the points 6; = (0,1/2,1). For the sensor location 5; = 0 or = 1 the estimation errors p 1 , 2 + 03. If the measurements are fulfilled in the middle of a specimen, = 1/2, then the inverse problem solution depends on the combination 6 1 + 6 2 of the desired temperatures and the roots p 3 , 4 become arbitrary magnitudes. Thus, in the class of constant initial-boundary conditions only three points of measurements do not provide the reconstruction both the model coefficients a 1 , 2 and boundary temperatures 2 1 1 ~ . Any other sensor location ensures the identification of thermal properties and boundary temperatures simultaneously. The further important problem here is a determination of optimal initial-boundary conditions and sensor location to provide the minimal errors p i - 4 for fixed S # 0. This problem will be studied in the future. Let us now consider other commonly known inverse problem with a heat flux loading scheme. We specify the following mathematical model

0): (8," - ~ ( a ~ ) ) u = ( to, ~in)

3 x R,:

where u = t(ul,u2,u3) is the displacement vector and L(&) = Cf,j=,aijd,; a,, . The coefficients aij are 3 x 3-matrices whose ( p ,4)components aipjq are given by aipjq = X S i p S j q 2p(6ijdPq 6iq6j,), where X and p are the Lame constants and Sij are Kronecker's delta. The density is assumed to equal 1. Plane waves in the whole space R3 mean the solutions of the form

+

ceiu(t--qz)v

(a > 0, c E

+

c),

where 77, v E R3 are taken to satisfy det(1- L ( q ) ) = 0 and v E Ker(1- L(7)). In the case of P-wave, the direction 77 of the propagation and the direction v of the amplitude are parallel each other, and in the case of S-wave they are perpendicular. In the half-space IR; we add some waves to the above plane wave (the incident wave) so that a boundary condition is satisfied. Here we impose the

184

Neumann boundary condition

Nu

=

c 3

viaija,,u [z3=o = 0,

i+l

where Y is outer unit normal vector to the boundary (i.e. v = ‘(O,O, -1)). The added waves are, so called, the reflected waves. We can classify the phenomena of reflection in the following way:

(P)

For an incident P-wave, P- and S-waves are reflected.

(SV)

For a n incident S-wave, P- and S-waves are reflected.

(SH)

For an incident S-wave, only S-wave is reflected.

(SVO) For a n incident S-wave, S-wave is reflected together with the evanescent wave.

(SVO) is the total reflection. Furthermore we have different wave not associated with the reflection (P) (SVO): N

(R)

There exists the wave called “the Rayleigh wave”.

The evanescent and Rayleigh waves are concentrated exponentially near the boundary, and called the surface waves. Because of the surface waves, formulation of the theory in R$ becomes different from that in R”,as is described in section 5. Getting rid of the part eiat from the above plane waves and the Rayleigh wave, we call the remainder functions “the generalized eigenfunctions” since they satisfy L(&)u = -u2u. Dermenjian and Guillot’ have shown that any data are expressed by superposition of these generalized eigenfunctions, and have developed the scattering theory of the Wilcox type for the equation in the half-space. We shall explain their results later (in section 3). 3

The Generalized Fourier Transformation

The Fourier transformation F is a powerful tool in the scattering theories. The Fourier transform F [ f ]= f^ of f is of the form f( 0,w

E

s,

where S, is the zone associated with the case a (U,S, 6;is the incident wave of the form

( a = P,

SV, SH, SVO

),

161 = l}),

= ( 6 E R;:

&(x; o , w ) = m , ( ~ , w ) e ~ ~ ~a “, ((w~) ) ”

(2) and Fa(%; 0 , w ) is the reflected wave for the incident wave 6;. Furthermore, m,(cr,w) is a function satisfying Im,(a,w)l = 1, and v a ( w ) , a,(w) are some vectors satisfying det(1 - L(va(w))= 0 and [I - L(q,(w))]a,(w) = 0 ( a = PI SV, SH, SVO). Let us note that all m, ( a = P, SV, SH, SVO) were taken equal to 1 in Dermenjian and Guillotl. The generalized eigenfunction 4 R of the Rayleigh wave is the form 2

4R(%;0, = C cjeio&(C)”aj~ ( w , < ) 7

< E sR7

j=1

.

.

where SR = {Q E R2;1(’1 = l}, cj are some constants and &, a; are some vectors satisfying det(I - L(&) = 0, (I- L ( q k ) ) a k = 0. In detail, see M.

186

Kawashita, W. Kawashita and Soga2. The third component q i 3 of qk is taken ; C) decays exponentially as z3 + +oo. satisfying Im[qi3] > 0, and so 4 ~ ( z(T, The generalized Fourier transformation (spectral representation) is defined in the following way:

F ( 0 ) = ( F P ( O ) ,Fsv((T),F S H ( @ ) ,.Tsvo(cJ),F R ( U ) ) , (FLY(a)f)(w) = CLY(-i(T)(f,4LY(~;(T,W))H, w E SLY,

(3)

where ca are some constants. The Fourier transformation plays an important role also on the LaxPhillips theory. Lax and Phillips expressed concretely the solutions in the free space by mean of the Radon transformation (cf. Lax and Phillips3):

This operator is much connected with the Fourier transformation: R f ( s ,0) = & eiso f ^ ( ~ w ) dTherefore, ~. it is expected that we can derive the various expressions in the Lax-Phillips theory from the results of the Wilcox type. In fact, this expectation is accomplished, and moreover both the settings (the Wilcox and Lax-Phillips types) can be changeable even in the abstract situations. But there are many choices on selection of the above superposition (i.e. selection of the functions & ( o , w ) ) , and each choice of &(o,w) is corresponding to one translation representation in the Lax-Phillips setting. In the Lax-Phillips theory we are required to choose the representation nicely to have a good property, and so every discussion is not finished by the one of Dermenjian and Guillot' . More precise discussion is given later in section 5. 4

Relation Between The Wilcox and Lax-Phillips Theories

In this section we consider the (abstract) wave equation

8

( 7 - L)u(t)= 0 ,

dt where -L is a positive self-adjoint operator on a Hilbert space 31. In the LaxPhillips3 scattering theory , one of the main assertions is that the solution operator

can be transformed into translation by some two operators T*: There exist subspaces D& in the space of the data (the energy space H ) and unitary

187

operators T* from H to L 2 ( R i ; N )( N is an extra Hilbert space) such that

if and only if there exist subspaces D* in H satisfying

(i) U(t)D* C D* for any f t

> 0,

(iii) UtcwU(t)D* are dense in H . T+ and T - are called the outgoing and incoming translation representations, and D+ and D- are called the outgoing and incoming subspaces. In the Lax-Phillips theory, the scattering operator S is defined by S = T+(T-)-l, and is desired to contain all the information about the scatterer. The generator of U ( t ) is of the form

A = ( -L I0 ) and the spectral representation for A means (in the Lax-Phillips sense) a unitary operator 7 from H to L2(R1;N ) such that

7 A = ia7 We see that this 7 is connected with a translation representation T by the equality: ( r f ) ( s )= J e - i u s ( T f ) ( s ) d s (f = t ( f i , f 2 ) E H ) . The spectral representation 7 in the Lax-Phillips sense can be translated into the one in the Wilcox sense F ( a ) (the generalized Fourier transformation in section 3): Theorem 1 (i) If we have 7 or F ( a ) , then by this we can make the other.

(ii) These 7 and F ( u ) are connected each other by the equality

This theorem is proved in M. Kawashita, W. Kawashita and Soga2.

188

5

Expressions in the Free Space

In this section we consider the isotropic elastic equation in the half-space IF$ and construct the fundamental expressions in the Lax-Phillips theory (e.g., the translation representations, etc.). In the whole space IW’” those are described by Lax and Phillips3 (for the d’Alembert equation), Shibata and Soga5 (for the elastic equation). The construction in the half-space is fairly different from that in the whole space. This is mainly due to existence of the surface waves (i.e., the Rayleigh and evanescent waves). If we want only to make a translation representation T (or spectral representation 7), then by Theorem 1 we can derive it soon from the generalized Fourier transformation F ( g ) (of the Wilcox type} which has been obtained by Dermenjian and Guillot’. In the Lax-Phillips theory, we like to construct T with a good property: Lax and Phillips made a translation representation T such that

[ U ( t ) f ] (= ~ )0 for all ( t ,z) with 1x1 5 t (> 0 ) if and only if ( T f ) ( s )= 0 for all s < 0.

(6)

This implies that D+ consists of the data to make the lacuna arise. Lax and Phillips3 carried out the construction of T with the property ( 6 ) very concretely by means of the Radon transformation (4). This concreteness and the property ( 6 ) are very useful for further investigations on scattering problems, e.g., the inverse problems (cf. Majda4, Soga6y7,etc.). Thus we hope that our translation representation also has the property ( 6 ) . As is explained later, however, we cannot get such a representation, and only can obtain the one with partially similar property (cf. Theorems 3 and 4). Let F ( c ) = (FP(~),FSV(~),-TSH(~),F~VO((T),.TR(~) be the generalized Fourier transformation (3) defined in section 3. Then, by (3) and (ii) of Theorem 1 we obtain the spectral representation 7 and consequently the translation representation T (in the odd dimensional half-space,:XE T - becomes equal to T+, i.e., T+ = T - = T ) :

Theorem 2 W e can obtain a translation representation T with the properties stated (5) which is of the f o r m T = (Tp,. . . ,TR)and is a unitary operator from H to L2(R1; N ) where N = @,E*L2(S,), A = { P , .. . , R}. We can express T by means of the Radon transformation in (4) also. But then we need to employ the following modified one f i j to deal with the surface

189

waves.

where 8 E SSVO(or SR) and ij(8) = (ij‘(8),ij3(8))is a certain vector satisfying det(1 - L(ij(8)))= 0. In detail, see $5 in M. Kawashita, W. Kawashita and Soga2. Since we can reconstruct the data f by F ( a ) and F(u)*(cf. (l)),we can express concretely the solution u(t,x) (or U ( t ) f )by means of the above translation representation T (cf. $5 of Kawashita et a1 2 ) . Noting this expression of U ( t ) f ,we can decompose U ( t )f into two parts:

U(t)f = U B ( t ) f -/- U S R ( t ) f , where U B ( t )f is superposition of the plane body waves associated with the real roots ij of det ( I - L(ij)) = 0 and U S R ( t ) f is the remainder term, i.e., consists of the surface waves. We choose the functions m , and 7, in ( 2 ) as follows: m,(a,w) = 1 for

= P, SV, SH

(Y

vlP(w)= cplt(w’,-w3),

,

~ , ( w ) = c;’ t ( w ’ , -w3)

for

(Y

= SH, SV,

svo

where cp and cs are the propagation speeds of the P- and S-waves respectively. Then we can see that T has a similar property to ( 6 ) , but weaker than (6):

Theorem 3 f E H belongs to D*,i.e.,

( T f ) ( s )= 0 f o r all f s

< 0,

if and only if the following conditions (a) and (zi) hold: (2)

SUPPP [ U B ( t ) f l l c { x E

(22) SUppP [ U ( t )f ] l l s s = O

@; fcst < Ixl},

c {d E

where Pt(x’,x3) = x‘ and cs,

CR

fCRt

< Ix’I},

are some constants independent o f f , t and

2.

It is because of existence of the surface waves to need to choose m, and 7, in the above way. In case of the half-space, we cannot make the translation representation T with the exactly same property that (6), which follows from

190

Theorem 4 f E H satisfies the condition

supp [ U ( t ) f ]c .{ E q ; c t < 1x1) f o r a constant

c

> 0 if and only if U S R ( t )f

(t > 0 )

= 0, i e . , Tsvof = 0 and TRf = 0.

Lax and Phillips gave a characterization of the value of ( T f ) ( s0) , at any fixed (s, 0) for any f in some class: We choose a certain straight line { x ( t ) } t E w in R", and then we have ( T f ) ( s , 0 = ) lim t ~ o o t ( " - 1 ) / 2 [ U ( t ) f ] 1 ( x ( tFor ) ) . our equation we obtain a similar result:

Theorem 5 For any fixed (s,0) E S, (a = P, SV, SH) set x,(t) = c,(s

+ t)e, e = t(O', -&),

where c p and CSV = C S H are the propagation speeds of the P- and S-waves respectively. Assume that ( T f )( s ,6) is sufficiently smooth and decreasing (as Is1 -+ w). Then we have (T, f ) ( s , 0 )= 47~:'~ t+-m lim t [ U ( t )f]2(x,(t)). a,(@

f o r a = P, SV, SH

,

where a, is the vector in (2).

-

The above Theorems 2 5 are proved by M. Kawashita, W. Kawashita and Soga2. For the proofs, see $5 of Kawashita et a1 '. Acknowledgments

Mishio Kawashita was Partially supported by Grant-in-Aid for Encouragement of Young Scientists A-09740085 from JSPS, Wakako Kawashita were Partially supported by Grant-in-Aid for Encouragement of Young Scientists A-13740090 from JSPS, and Hideo Soga was Partially supported by Grantin-Aid for Sci. Research (C) 13640150 from JSPS. References

1. Y. Dermenjian and J. Guillot, Scattering of elastic waves in a perturbed isotropic half space with a free boundary. The limiting absorption principle, Math. Meth. Appl. Sci. 10, 87-124 (1988). 2. M. Kawashita, W. Kawashita and H. Soga, Relation between scattering theories of the Wilcox and Lax-Phillips types and a concrete construction of the translation representation, submmited.

191

3. P. D. Lax and R.. S. Phillips, Scattering theory (Academic Press, New York, 1967). 4. A. Majda, A representation formula for the scattering operator and the inverse problem for arbitrary bodies, Comm. Pure Appl. Math. 30, 165-194 (1977). 5. Y. Shibata and H. Soga, Scattering theory jor the elastic wave equation, Publ. RIMS Kyoto Univ. 25 , 861-887 (1989). 6. H. Soga, Singularities of the scattering kernel for convex obstacles, J.Math. Kyoto Univ. 22, 729-765 (1983). 7. H. Soga, Representation of the scattering kernel for the elastic wave equation and singularities of the back-scattering, Osaka J . Math. 29, 809-836 (1992). 8. C. H. Wilcox, Scattering Theory for the d’Alembert Equation in Exterior Domains, Lect. Notes in Math. 442 (Springer, Berlin, 1975).

FORMULAS FOR RECONSTRUCTING CONDUCTIVITY AND ITS NORMAL DERIVATIVE AT THE BOUNDARY FROM THE LOCALIZED DIRICHLET TO NEUMANN MAP GEN NAKAMURA Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo 060-081 0, Japan E-mail: [email protected]. ac.jp

KAZUMI TANUMA Department of Mathematics, Faculty of Engineering, Gunma University, Kiryu 376-851 5, Japan E-mail: [email protected] We consider the problem of determining conductivity of the medium from the measurements of the electric potential on the boundary and the corresponding current flux across the boundary, that is, from the Dirichlet to Neumann map. We give three kinds of formulas for reconstructing conductivity and its normal derivative from the localized Dirichlet to Neumann map. They are the formulas for pointwise reconstruction, reconstruction in a weak form, and reconstruction in the form of Fourier transform. In particular, the normal derivative of the conductivity at the boundary is reconstructed directly from the localized Dirichlet to Neumann map.

1

Introduction

Let R E R" ( n 2 2) be a bounded domain with Lipschitz boundary dR. Physically R is considered as an isotropic, static and conductive medium with conductivity y E L"(R). When an electric potential f E H1l2(dR)is applied to the boundary d o , the potential u solves the Dirichlet problem

V . (yVu)= 0 in R ,

ulan = f.

(1)

Assume that there is a constant 6 > 0 such that y(z) 2 b (a.e. z E R). Then, there exists a unique weak solution u E H1(R) t o (1). Define the Dirichlet to Neumann map A, : H1/2(dR) H-1/2(dR) by

-

(2)

where u is the solution to (l),2, is any function in H1(R) satisfying vlan = g and < , > is the bilinear pairing between H1l2(dR)and H-l12(dR). Note that A, f = yVu . n when f E H3/2(dR),y E C1(a) and dR is C2, where 192

193

n is the unit outer normal t o dR. Hence A,f is the current flux across dR produced by the potential f on dR. The problem of determining conductivity of the medium from the measurements of the electric potential on the boundary and the corresponding current flux across the boundary is expressed as I n v e r s e Problem “Determine y(z) from AY”. Since this problem was posed by A.P.Calderon, many results on uniqueness, stability, reconstruction have been proved. Here we give a brief review of some of the previous works on reconstruction. When y and dR are C”, using the fact that A, is a pseudodifferential operator in this case, Sylvester and Uhlmanng showed how t o recover y and all of its derivatives on 8R from the symbol of A,. When dR is Lipschitz continuous, from A, Nachman3 recovered y on dR if y E WIJ’(R) with p > n and recovered the first normal derivative of y on dR if y E W2+(R) with p > n/2. On the other hand, reconstruction of conductivity from the localized Dirichlet to Neumann map has been studied first by Brown’. Reconstruction from the localized Dirichlet t o Neumann map means that for zo E dR, assuming some regularity conditions on dR and on the conductivity locally around $0, we take Dirichlet data f’s t o be the functions compactly supported in a neighborhood of xo on d o , measure Neumann data A, f in that neighborhood and then reconstruct conductivity and its derivatives in that neighborhood. Under the condition that dR is C 2 and Vy is continuous locally around xo E d R , Brown’ reconstructed y and its first derivatives at xo from the localized Dirichlet to Neumann map. Recently, Nakamura and Tanuma4 reconstructed the higher order derivatives of y a t xo E dR i n d u c t i v e l y according to the regularity which y and dR have around 20. In this report, we give three kinds of formulas for reconstructing conductivity and its normal derivative from the localized Dirichlet to Neumann map. They are the formulas for pointwise reconstruction, reconstruction in a weak form, and reconstruction in the form of Fourier transform. Our standpoint is to reconstruct normal derivative of the conductivity directly from the localized Dirichlet t o Neumann map. More precisely, when recovering the normal derivative of y at 50 E an, we need only some regularity assumption on y around xo and need not any information on the values of y around XO. This standpoint is different from the reconstruction methods in Brown’, Nachman3 and Nakamura e t a1 4 , where they reconstructed conductivity and its normal derivative inductively. For example, when recovering the normal derivative of y at zothey needed t o know not only the value y(z0) but also all the values of y in a neighborhood of z0on dR in advance. (There is a recent work2 on inductive reconstruction by using only the value at 20.)

194

Our direct reconstruction of the normal derivative of y can be done by using two special kinds of Dirichlet data compactly supported in a neighborhood of xo on dR and A,. Full proofs are given by Nakamura and Tanuma'. In this report we give a brief sketch of the idea for these proofs. We believe that our direct reconstruction formulas are useful also for numerical computations. Finally we note that there are results on reconstruction of elastic tensor for the isotropic and anisotropic elasticity from the localized Dirichlet to Neumann map (Robertson*, Nakamura et a1 7). In Section 2, we give a formula which reconstructs conductivity and its normal derivative pointwisely at xo and give formulas which reconstruct them as the functions defined in a neighborhood of xo (in a weak form and in the form of Fourier transform). Since our reconstruction formulas involve limiting process, we give the estimates for their convergences in Section 3. 2

Reconstruction Formulas

To make the essential point of the problem clear, let us assume that dR is flat around x = 0 E d R and that R, dR are given by

a = { x , > O},

d R = { x , = O}

locally around x = 0, where x = (z',~,) = (zI,-..,x,-~,z,). Let t = (t',0 ) = (tl,.. . ,t,-1 ,0 ) be any unit tangent t o d R at x = 0. The starting point is the following theorem.

Theorem 1 (Brown') Suppose that y ( x ) is continuous around x = 0. Letting ~ ( x 'E) CF(R*-l) satisfy

we take 4N(x') = e

QNX'.t'

1 7 W X ' )

(4)

for any positive integer N . Then

Note: In Brown' he obtained this formula for more general class of y which includes piecewise continuous y and y in W 1 > l ( R ) .

195

For the formulas which reconstruct y and its derivatives inductively we have referred to Nachman3, Brown', Nakamura et a1 '. Here we give the formula in Nakamura et a1 4 .

Theorem 2 Let x ( x ) E C r ( R n )satisfy 0 5 x 5 1 on (1x1 5 E } , X ( Z ) = 1 and suppx C (1x1 < 2 ~ for } small E > 0. Define " k ( 2 ) ( k = 0 , 1 , 2 , . - . ) by k

yk = 1- x ( Z )

+ x ( x )(y(Z', 0) + XndZny(Z', 0) + . . . + 2k!,djck,Y(Z',

1

0) .

Assume that for k 2 1, d$dg;y is continuous around x = 0 for any multiindex (a',a n ) such that la'l+ 2 a n 5 2 k and let #N(z') be given b y (4). Then,

If xo can be any point in a small open subset I? of dfl, we can recover y on I? by using Theorem 1 and hence A,,, can be defined. Then, we can recover on r by using Theorem 2 with k = 1 and hence A,, can be defined. Repeating this process, we can obtain the higher order normal derivatives of y on r. In this sense the formula in Theorem 2 is an inductive reconstruction formula. Now we give our direct reconstruction formulas, which are the main results in this report.

2

Theorem 3 (Pointwise Reconstruction (Nakamura and Tanurnas)). Suppose that D$D,",-y is continuous around x = 0 for any multi-index (a',a,) such ) that la'[ 2an 5 2 . Letting ~(x')E C;(Rn-') satisfy (3) we take ~ N ( X ' in (4) and

+

Then,

In this formula, the left hand side is observable. On the other hand, the factors JRn-l(IVqI2 - (t' . 0 ~ ) dx' ~and ) (t' .w ' ) in the right hand sides, are controllable (except the case n = 2 ) , that is, these factors are determined explicitly from the Dirichlet data. Then from (6) we obtain a 2 x 2 system

196

of equations which can be solved for y(0) and z(0simultaneously. ) When n = 2 the factor &n-l (IVV~’- (t’ .VV)’) dx’ vanishes. So in this case we are able to reconstruct ( 0 ) immediately. Hereafter we assume that D$Dg;y is continuous around x = 0 for any multi-index (a’,a,) such that Ia‘I + a n 5 3, a, 5 1. Also we take ~ ( 5 ’to) be any function in C;(Rn-‘) compactly supported in a neighborhood of x’ = 0 and put

z

Theorem 4 (Reconstruction in a Weak Form).

Theorem 5 (Reconstruction of Fourier transform of conductivity). Let w’E R”-1 . Then

+(t’ . w ’ )

Ln-l

0 ) ~ ‘ ( x e\/=Tx”W’dx’. ’)

In (8), for given w’E Rn-’ we may take t’ E RnP1so that t’. w‘ = 0 ( n > 2). Then we get the Fourier transform of the normal derivative of y cut off around x’ = 0.

Outline of Proof : We briefly sketch the idea for the proof of Theorem 4. This idea can be applied also to the proof of Theorem 5. Full details are given by Nakamura and Tanuma‘. From the definition (2) we can write

197

where U N E H'(S2) satisfies ~ ~ l =a 4 n ~

V , .( ~ V U = N 0) in 0, and

@N

is an H 1 ( R )extension of

(9)

4~ of the form

= e- N x ' . t f e - N x ,

@N(x)

,

q(xl).

Note that this @ N is the first term of an asymptotic solution to (9), because the leading term of V@Nfor large N becomes

and from (10) we get N

S,

y(z) 2Ne-2Nxnq2(x')dx

for large N . Now in this integrant, the sequence

{2 N e - 2 N x n } m

N=l

converges to 6,,,20, " the delta function on the half line x , 2 0 ", as N in the sense that

as N

-+

-+ +oo

+oo for any a ( x , ) E Co([O,co)). Therefore we get

y ( x ) 2Ne-2Nxnq2(x')dx

+

Ln-l

y(z',0) q2(x') dx'

( N + +m),

which proves (i). We have used a sequence converging to 6,,20, which enable us to extract the value of y ( x ) at x , = 0. So, for (ii), we first propose taking a sequence which is obtained by differentiating each term of the sequence ( 1 1 ) : d { -dxn 2Ne-2Nxn)

oc,

N=l .

198

This sequence converges to LL the derivative of 6>,0 ", and using this we may expect to extract the xn-derivative of y(x) at x, = 0. a However, we get from the integration by parts

I"

1

2Ne-2Nxn C Y ( Xdxn ~ ) = ~NcY(O) -t

2 N e - 2 N x n ~ ' ( ~dxn. n)

Although the second term tends to a'(0) as N 3 +00, we must have the first term which goes to infinity. This is because S,,~O is not a usual delta function defined on the whole line -cm < x, < 00. This implies that we should first take a sequence converging to 6,,>0, each term of which vanishes at xn = 0 and then make a new sequence by differentiating each term of that sequence. Like the proof (12) for 2Ne-2Nxn + 6,n>o ( N + +00), we easily see that 2Ne-Nx"

+ 26,,20

(N

--+

+m).

Since 2Ne-2Nxn - 2Ne-Nx- vanishes at xn = 0, the sequence

is a desired one. In fact,

Therefore 2N2e-Nxn - 4N2e-2Nxn)v2(x')dx

-+

-(x',O)

q2(x') dx'

(13)

( N --+ +00).

Thus, if we choose ! € J N ( z ) = eG

T

N " - X

.t

e cxnq(x'),

the first term of an asymptotic solution to

=Recall that as a linear functional the first derivative of the delta function maps the test function to the minus of its first derivative at the origin.

199

then we get for large N

Since

we see that the dominant term of the last integral (15) for large N is given by the left hand side of (13). 3

Estimate of Convergence

When we assume higher order regularity on y around x = 0, we can get the estimates for the convergences in the formulas of the previous section. As examples, we give the estimates for the formula in Theorem 3 and the formula (ii) of Theorem 4. The proof is given by Nakamura and Tanuma6. Theorem 6 (i) Suppose that D$Dg;y is continuous around x = 0 for any muZti-index (a',a,) such that la'l 2a, 5 4. Letting q ( x ' ) E C,"(Rn-l) satisfy (3), we take ~ N ( x 'and ) + N ( x ' ) in (4) and (5) respectively. Then there exists a constant C which depends on the values d,S'd:;y (la'] 2a, 5 4) in a neighborhood of x = 0 such that

+

+

bMore precisely, + N and * N should be the summations up to the second terms of the asymptotic solutions to (9) and (14) respectively. However, it can be proved that these second terms do not have any effect on the leading term of the integral (15) (see Nakamura and Tanuma 6 ) .

200

(ii) Suppose that D$D,“,-r i s continuous around x = 0 f o r any multi-index (a’,a,) such that Ia’I + a, 5 4 ,a, 5 2. Letting ~ ( x ’ be ) any function in C$(Rn-l), we take q5~(x’)and $ J N ( x ’ )in (7). T h e n there exists a constant C which depends o n the values 8Et’8z;y (la’] a, 5 4, a, 5 2: in a neighborhood of x = 0 such that

+

Acknowledgments The first author is partly supported by Grant-in-Aid for Scientific Research (B) (No. 14340038), Society for the Promotion of Science, Japan. The second author is partly supported by Grant-in-Aid for Scientific Research (C) (No. 13640115), Society for the Promotion of Science, Japan. References 1. R. M. Brown, Recovering the conductivity at the boundary f r o m the Dirichlet to N e u m a n n map: a pointwise result, J . Inverse and Ill-posed Prob. 9(6), 567-574 (2001). 2. H. Kang and K. Yun, Boundary determination of conductivities and Riemannian metrics via local Dirichlet-to-Neumann operator, preprint. 3. A. I. Nachman, Global uniqueness f o r a two dimensional inverse boundary value problem, Ann. of Math. 142, 71-96(1995). 4. G. Nakamura and K. Tanuma, Local determination of conductivity at the boundary f r o m Dirichlet t o N e u m a n n map, Inverse Problems 17, 405-419 (2001). 5. G. Nakamura and K. Tanuma, Direct determination of the derivatives of conductivity at the boundary f r o m the localized Dirichlet t o N e u m a n n map, Comm. Korean Math. SOC.16, 415-425(2001). 6. G. Nakamura and K. Tanuma, Reconstruction of conductivity and its normal derivative at the boundary f r o m the localized Dirichlet t o N e u m a n n map ,preprint. 7. G. Nakamura and K. Tanuma, Reconstruction of elastic tensor of anisotropic elasticity at the boundary f r o m the localized Dirichlet t o Neum a n n m a p , preprint.

20 1

8. R. L. Robertson, Boundary identifiability of residual stress via the Dirichlet t o N e u m a n n map, Inverse Problems 13, 1107-1119 (1997). 9. J. Sylvester and G. Uhlmann, Inverse boundary value problem at the boundary-continuous dependence, Comm. Pure Appl. Math. 61, 197219 (1988).

HOCHSTADT-LIEBERMAN TYPE THEOREM FOR A NONSYMMETRIC SYSTEM OF FIRST-ORDER ORDINARY DIFFERENTIAL OPERATORS IGOR TROOSHIN Institute for Problems of Precision Mechanics and Control Russian Academy of Sciences, Saratov, Russia E-mail:[email protected]. or.jp MASAHIRO YAMAMOTO Department of Mathematical Sciences, The University of Tokyo 3-8-1 Komaba, Meguro, Tokyo 153 Japan E-mail:[email protected]. ac.jp We consider an eigenvalue problem for a nonsymmetric first order differential operator A u ( z ) =

( y i) g(z)+

Q ( z ) u ( z ) ,0

< z < 1, where

Q is a 2 x 2 matrix

whose components are of C' class on [0,1]. Assuming that Q(z) is known in the half interval of (0,l), we prove the uniqueness in an inverse eigenvalue problem of determining Q ( z ) from the spectra.

1

Introduction and the Main Result

We consider a non-symmetric first-order differential operator A Q ,J~ ,in {L2(0,l)}? du

(AQ,j,Ju>(x) = B -dx (x)

w(0) +ju,(O) = 0,

Q(2)42),

uz(1)

u E D(AQ,~,J),

+ JUl(1) = 0 } .

Similarly we can define operators A Q , ~ ,,JAp,h,H, * Ap,h,H*, etc. where We assume that P = ( P k e ) l < k , e < Z , Q = (qke)l

where C = C(t7 a=a(t, and b=b(t7 are complex functions o f t 0 and E ( - 0 0 , CQ). Moreover we assume that the functions C , a and b are chosen so that the right-hand side of equation (9) determines the function absolutely integrable over 2 along the whole real axis. One can easily verify that the requirement will certainly be satisfied if the function E and r of the form as argued in Mel'nikov2

c

d d E = Ic(t7 C)l[b(tl c)l+b(tl c)1l27r= Ix[c(tl c>a2(t7c)ll+lz[c(h 0b2(t7

c)]/

215

at any t 2 0 satisfy the condition

3

The Lax Representation

41,z

=-x41

+442,

427

= 441

+ iC42

C E (-WOO)

According to (4),(8) and ( l l ) , we may define

-

6 2. -- a .2, b2. -- b2. ,-c i = c i ,

62n+2m+1

Then

where 8 is some constant and

= 0,

i=0,1,-'.,2n,

m = 0,1,. . . ,

(1lb)

216

also satisfies the adjoint representation (2), i.e. @n+')

= [U,j$7(2"+1)],

(12)

which, in fact, gives rise to the Lax representation of (11). Since (11) is the stationary equation of (9), it is easy to find that the zero-curvature representation for the mKdV hierarchy with integral type of source (9) is given by Utz,,, - j q n + l ) + [U,j p n + q = 0, (13) with the auxiliary linear probIems

where X = iC and

~ h , t ~ , , (+5, ,t ~ n + ,l

C) = C ( 2 n + l )$1 + (

+ O)$a

(14b)

In this way we find the explicit evolution equations of eigenfunction $. Indeed, this kind of evolution equation of eigenfunction was not obtained in Mel'nikov2r3.

217

4

Evolution Equation for the Reflection Coefficients

We define the eigenfunctions f-(z, C) = (fF(z,C), fT(z, f-(z, C) = (fJGC)7faz,C))Tl f+(z,C) = (fl+(.,o,f2+(.,C))T and f+(z,C) = (f:(x, C), f:(z, C ) ) T for the equation (14a), and the following asymptotics are fulfilled at any E ( -cm, cm)


0, and the functions f-(z7C) and f+(z,C) admit an analytical continuation in the parameter C into the lower half-plane ImC < 0. It is easily seen that at any real C E (-00,cm) the pair of functions f-(z, r,

(28) W e also (29)

Analogues t o theorem 1 we can establish that Theorem 2 Suppose the condition (28)-(30) hold, let us select

where the sense of bracket [u] is the same with the theorem 1, then we have estimate

where C is a constant.

M M(1n -)-2(s-') &

+ 0,

as E

+ O+.

So the conclusion of theorem 2 is really an improvement of theorem 1 near x = 0.

246

Acknowledgments

The project is supported by the Natural Science Foundation of Gansu province (ZS021-A25-001-Z) and the National Natural Science Foundation of China (No.49875024). References 1. J. V. Beck, B. Blackwell and S. R. Clair, Inverse Heat Conduction: IllPosed Problems (Wiley, NewYork, 1985). 2. A. Carasso, Determining surface temperatures from interior observations, SIAM J. Appl. Math. 42(3), 558-574 (1982). 3. T. Regiriska, Sideways heat equation and wavelets, J. Comput. Appl. Math. 63, 209-214 (1995). 4. C. L. Fu, C. Y. Qiu and Y. B. Zhu, A note o n "Sideways heat equation and wavelets" and constant e*, Comp & Math with Appl. 43(8/9), 1125-1 134 (2002). 5. L.EldQn, F.Berntsson and T.Regiriska, Wavelet and Fourier methods for solving the sideways heat equation, SIAM J. Sci. Comp. 21(16), 21872205 (2000). 6. I.Daubechies, Ten Lectures on Wavelets (SIAM,Philadelphia, 1992). 7. Dinh Nho HBo, A. Schneider and H-J Reinhardt, Regularization of a noncharacteristic Cauchy problem for a parabolic equation, Inverse Problems 11, 1247-1263 (1995).

DIRECT SIMULATION OF AN INTEGRAL EQUATION OF THE FIRST KIND H. IMAI Department of Applied Physics and Mathematics, Faculty of Engineering, .University of Tokushima, Tokushima 770-8506, Japan E-mail: [email protected] T. TAKEUCHI Department of Applied Physics and Mathematics, Faculty of Engineering, University of Tokushima, Tokushima 770-8506, Japan E-mail: [email protected] Direct numerical simulation to an integral equation of the first kind is carried out by using IPNS(1nfinite-Precision Numerical Simulation). Numerical results are very satisfactory in accuracy. Moreover, they also show some interesting facts. These numerical results show IPNS facilitates numerical analysis for such inverse problems.

1

Introduction

Inverse problems are very difficult t o be solved. Numerical simulation is inevitable in practical analysis. Many mathematical problems are analyzed by using direct numerical simulation. However, it has been a taboo t o inverse problems due to easy corruption by strong oscillation. For avoidance of this oscillation some additional methods are usually used together. In such methods original inverse problems are often transformed into modified problems, then solved. Here we should remark this modification is important from the practical view point, however it is not preferable from the view point of analysis. These additional methods are the regularization, the method of least squares and AI. Restriction of the dimension of the solution space is very popular in concrete analysis, and it is a sort of the regularization. A1 facilitates the development of the efficient solver of the problem and the implementation of experiences t o the solver. Unfortunately these additional methods are not absolute. This is because in numerical simulation the rounding error spoils their theoretical usefulness. As for direct numerical simulation of inverse problems some new approaches were carried out. Multiple-precision arithmetic is a keyword. It removes the effect of the rounding error t o strong oscillation. Multiple-precision arithmetic was applied t o the following integral equation of the first kind. 247

248

Problem 1 Find u(y) such that

The exact solution for Problem 1 is u(y) = y. Direct numerical simulation was carried out. Numerical results in multiple precision were satisfactory comparing with those in double precision2. On the other hand, IPNS(1nfinitePrecision Numerical Simulation)' was applied to several inverse problems governed by P D F system^^^^^^^^. It was also applied t o Problem 1 '. Numerical results were very satisfactory, however numerical investigation was not in detail. In the paper more numerical investigations are carried out. 2

Application of IPNS

2.1 Infinite-Precision Numerical Simulation Numerical errors originate from the truncation error in the discretization and the rounding error. Realization of highly accurate numerical simulation needs arbitrary reduction of both errors. For such numerical simulation we proposed a simple method called IPNS(1nfinite-Precision Numerical S i m ~ l a t i o n ) ~ .IPNS consists of the arbitrary order approximation and multiple-precision arithmetic. The former is used for the arbitrary reduction of truncation errors in the discretization. The last is used for the arbitrary reduction of rounding errors. For the arbitrary order approximation spectral methods are very useful'. Especially, the spectral collocation method is most useful. Its application is same as FDM, so it is easily applicable to nonlinear problems, even t o free boundary problems. In the spectral collocation method, the order of approximation can be controlled by the number of collocation points.The multiple-precision arithmetic is now easily available. A lot of FORTRAN subroutines about it are already prepared. Some libraries are free and distributed on the net, e.g. http ://www .lmu. edu/acad/personal/f aculty/dmsmith2/FMLIB. html'. IPNS has been applied t o many problems and ultimately high accuracy has been seen in numerical results.

2.2

The Way of Application

IPNS was applied to Problem 1 as follows4. The problem should be transformed t o be defined in the interval [-1,1]. So, we consider the following

249

problem.

Problem 2 Find u(y) such that

The exact solution for Problem 2 is

Remark 1 This problem is derived f r o m Problem 1 as follows. From the

',

transformation y = - Problem 1 becomes 2 +

X

Taking - = t and u 2

(q) = v ( t ) , then

Then it is easy t o see Problem 2 is derived from this. Such transformation is necessary f o r application of Chebyshev polynomials, however it is not restriction. Remark 2 Problem 2 i s a- 1 special case of the following general case:

To Problem 2, the spectral collocation method with Chebyshev polynomials is applied to the integrand as follows4: N

e"Yu(y)

C k=O

From the inversion formula

(Y).

~k ( x ) ~ k

(5)

250

where j7r

j = 0, 1;.-

Y~=COS--,

N

uj :

cj =

{yj}

, N,

the computed value of u ( y j ) , 1,

j = 1, 2 , . . . , N -1,

2,

j = 0, N .

(7) j = 0, l,... , N ,

(8) (9)

are called C-G-L(Chebyshev-Gauss-Lobatto) points1. Thus

2 N

=-

1 C c =exYju N

N

k=O k#l

j=O

jkr l+(-l)k cos - . N 1-k2

'

(14)

We choose proper points { x L } , I = 0, 1,.--, N on which equation is satisfied. Set f i = f ( z L )then , we have the following linear system:

where

After solving this linear system, u(y) is reconstructed as follows : N

N

Remark 3 The same discretization can be carried out t o the general case in Remark 2. Remark 4 Eqs. (5) and (6) are not inversion formulae f o r u(y) except f o r the case x = 0. This means this reconstruction is not obvious. In the general case in Remark 2 this exceptional case is that k(x,y) is independent of y. Remark 5 Our discretization is done not to u(y) but t o exYu(y). This is because it is applicable t o the general case in Remark 2 without numerical integration which generates the additional truncation error. However, we are not sure that our discretization is best. The above linear system (15) is very ill-conditioned, so numerical computations must be carried out in multiple-precision. This is IPNS.

3

Numerical Results IT

Figure 1 shows errors for Problem 2 with C-G-L points : XI = cos -, N 0,1,. . . ,N. Here, error= max I u ~ ( y j ) - u ( y j ) l , yj=cos-,+,r OSjSN N

j=O,l,...,N.

1=

(18)

u(y) is the exact solution, and u ~ ( y )is the right-hand side of Eq. (17). Here we should remark ~ ~ ( y =j )u j . If the rounding error is not small enough, the error defined by Eq. (18) grows explosively before obtaining good results. This shows the linear system (15) is very ill-conditioned. At the same time, if the rounding error is small enough, the error reduces successively. In Figure l(d) the regression line by the method of least squares is log (error) =

(b) Quadruple precision

(a) Double precision

(c) 1000 digits

(d) 2000 digits

Figure 1. Behavior of maximum errors for Problem 2.

-3.00 * log N - 0.686 with the correlation coefficient p = -1.00. This means IPNS works well. Figure 2 shows error dependence on the choice of

(400 digits). Error by using C-G-L points :

(21) for

ZIT

21

= cos -,

N

Problem 2

1 = 0,1, . . . ,N

is compared with error by using equally spaced points in [-1,1] :

+

XI

=

21

-1 -, I = 0,1,. . . ,N . There is almost no difference in the behavior of N error reduction.

Figure 3 shows error dependence on the interval where {xl} are distributed. Equally spaced points in [-lovm, : xl =

lo-"

(-1

+ $) ,

1 = 0,1,. . . ,N are used for obtaining the linear sys-

253 1e+OM

-C-G-L ........

1e+006

points+

Equally spaced points'

x2

10000

100

1

0.01

0.0001

1 e-006

le-OM I 10

20

40

Figure 2 . Error dependence on the choice of

60

(21)

80

100

160

200

for Problem 2(400 digits).

tem (15). Behavior of errors is quite different whether N is odd or not. For even N spectral accuracy is seen. Remark 6 Remark Figure 3.

4

4

m a y mean the appearance of spectral accuracy seen in

Conclusion

Direct numerical simulation to an integral equation of the first kind is carried out by using IPNS. Numerical results are very satisfactory in accuracy. Moreover, they also show some interesting facts. IPNS sometimes needs long CPU time and huge memory space because it involves multiple-precision arithmetic. However, numerical computation with several hundreds digits is already practical. IPNS facilitates numerical analysis for inverse problems. Direct simulation t o inverse problems is not a taboo now.

254 le+010

1

le-010

1 e-020 L

0

b 1 e-030

le-040

1 e-050

1 e-060

Figure 3. Error dependence on the interval including {zi}(400 digits).

Acknowledgments

This work is partially supported by Grants-in-Aids for Scientific Research (No. 13640119), from the Japan Society of Promotion of Science. References

1. C. Canuto et al., Spectral Methods in Fluid Dynamics (Springer, New York, 1998). 2. H. Fujiwara and Y. Iso, Numerical Challenge to Ill-posed Problems by Fast Multiple-Precision System, in Proc. the 50th Japan National Congress on Theoretical and Applied Mechanics , 419 (2001). 3. H. Imai and T. Takeuchi, Application of the Infinite-Precision Numerical Simulation to an Inverse Problem, NIFS-PROC 40, 38-47( 1999). 4. H. Imai and T. Takeuchi, Some Advanced Applications of the Spectral Collocation Method, GAKUTO Int. Ser. Math. Sci. Appl. 17,323(2001).

INVERSE PROBLEM OF RECONSTRUCTING THE PARABOLIC EQUATION’S INITIAL VALUE AND THE HEAT RADIATIVE COEFFICIENT * YONGJI TAN Department of Mathematics, fudan university, Shanghai, China E-mail: [email protected]

CHUNXIA JIA Department of Mathematics, Shanghai Normal University, Shanghai, China E-mail: [email protected] In this paper, the numerical method to reconstruct the radiative coefficient and initial condition simultaneously by measuring the domain temperature at a fixed time and the temperature of a subdomain all the time is studied. By least-square technique this inverse problem can be formulated into a variational problem and discretized into nonlinear programming problem with the cost function depending on the numerical solution of the corresponding direct problem of heat equation. We obtain the numerical solution of the direct problem by finite difference method and radial basis function (RBF)method respectively and derive the gradient formula for cost function, then implement the numerical reconstruction by quasi-Newton technique. In the case of the measuring data with noise, we use regularization method. Numerical results show that this method is available.

1

Introduction

Consider the following initial-boundary problem with the radiative coefficient: dU = Au + p ( z ) u , in R x (O,T) at

u(z,O) = p(z), in R u ( z ,t ) = ~ ( zt ), , in dR x ( 0 , T )

(1) (2) (3)

where u = u ( z ,t ) is an unknown temperature function, p ( z ) is radiative coefficient, the physical domain R is an open bounded domain in Rd(d = 1 , 2 , 3 ) , with a piecewise smooth boundary do. In this paper, we mainly investigate the numerical method for reconstructing the initial temperature distribution p ( z ) and the heat radiative coefficient p ( z ) in (1) - -(2). It is well known that given only the measurement of temperature at a fixed time T(> 0 ) , the reconstruction of the initial temperature *PROJECT 10171020 SUPPORTED BY NSFC.

255

256 is highly ill-posed, let alone the case that we intend to recover both initial temperature and radiative coefficient here. Having some extra observation of the temperature, say in a small subregion of the physical domain w along the time direction, it is possible to reconstruct p(x)and p ( x ) . M. Yamamoto etc. proved that the problem is conditionly stability, by formulating it into a variational problem, and using finite element discretization and gradient method, they achieved the numerical reconstruction. Let $ T ( x ) be the measurement of u ( z , T ) , $(x,t) be the measurement of u ( z , t )in w x ( O , T ) , the problem of reconstructing p ( z ) and p ( x ) can be formulated into that to find (p(x),p(x),~(z, t ) )which satisfies (l),(2), (3) and the following equations:

By least-square method, the problem can be formulated into minimizing the following cost function with constrained conditions(1)-(3)

In this paper, for given discrete p(x) and p ( x ) , we will solve boundary value problem (1)-(3) by finite difference method and RBF method respectively and therefore obtain the discrete value of Q ( p , p ) . After deriving the gradient formula for Q ( p , p ) , we implement the reconstruction of p and p by quasi-Newton technique. In the case of the existing measuring errors, we use regularization method with both regularization terms

and

where the constant CY and ,B are regularization parameters, p, and p , are guesses of p and p respectively. 2

Minimizing the cost function by quasi-Newton method

Consider approximate values pi = p ( x i ) and pi = p ( x i ) of p(x) and p ( x ) at , x Given ~ } p i , pi(i = 1 , 2 , - . - , N ) we , discrete point sets { x ~ , x 2 , ~ ~in~ 0.

257

can numerically obtain the temperature value u(q,t j ) at (xi,t j ) , and then find the approximate value of the functional Q ( p ,p ) , therefore Q ( p , p ) can be approximately expressed as Q ( p l , p 2 , . . . ,prv, p1 ,112, . . . ,prv). Let

P = ( P l , P Z , . . . , P N , P l , 1.12,. . .,PN)' the cost function can be written as

Q = Q(P)

(7)

In quasi-Newton method an iteration sequence pol p1 , . . . is designed to approximate the minimum of Q ( p ) . The descending direction after kth iteration is:

dk = -[V2Q(pk)]-1VQ(pk)

(8)

where V 2 Q ( p k ) is Hesse matrix Q ( p ) evaluated at p = p k . We use H k t o approximate V 2 Q ( p k ) ) which , is obtained by updating BFGS formula:

where sk

= pk+' - p k ,

Yk

= VQ(pk+') - V Q ( p k )

Therefore, the main process of quasi-Newton method is as follows:

1. initialization: select proper initial point po E R N , let H o = I , k = 0, calculate Q ( p o ) and V Q ( p o ) ; 2. compute the fastest descending direction: dk = - H k V Q ( p k ) ;

3. one dimensional search: solving one dimensional optimization problem minQ(pk t>O

to obtain t = t k llet pk+' = p k

+t d k )

+tkdk;

4. updating matrix H k : using BFGS formula t o update H k , get H k + l ;

5. compute V Q ( p k + ' ) , if IlVQ(pkf')ll erwise, goto 2

< E (a given small value), stop; oth-

From above, we know the main job is to calculate Q ( p k ) and V Q ( p k ) , especially the calculation of V Q ( p k ) is troublesome. If the finite element method is used to solve the direct problem, sensitivity coefficient method and adjoint state method are often used to calculate the gradient, where many linear algebraic equation systems should be solved, nevertheless, if we use difference method or RBF, it is possible to obtain the explicit expression of the gradient.

3

the calculation of cost function and its gradient

3.1 RBF method

For a given function g(z), we take =9(11X-Xjll>

9j(.)

j =l,-..,N

as radial basis, where 2 1 , . . . , XN are given grids. Let N

u ( z ,t ) = c a j (t)9j> .( j=1

dU Un+l-u; Denoting u(z2,n A t ) by u;,-(z, nAt) can be discretized into at At is time increment . (l)-(3)can be written as: N

N

, where

259

By (13) - -(15),we can determine 07, and furthermore by ( 1 2 ) we can deduce The matric form of (13) - -(15) can be written as

@'.

AD" = R"

(16)

where an = (O;",a;,. . ,a%)

the matrix form of ( 1 2 ) is

Un+l = @.t.B.an+G.A.cu" where:

U" = (u?,u;, . . . u;)' uy denotes the value of u at the jth grid point after n iterations. A and B are matrices depend only on the grids coordinate, G is a diagonal matrix. It is easy to see that we can obtain the derivative of Un+' with respect to the parameter and obtain V & ( p ) . 3.2

Finite difference method

For convenience, we discuss the one dimensional case and assume that R = (0,l). In this case, boundary condition ( 3 ) can be written as

and the difference scheme of (1) is U?+' 3

- U?

- Ujn+'

3 -

h

- 2ujn k2

+ ujn-l + PZLp

therefore we have

u;+' = Auy-,

whereA=-,

h

k2

+ Buy + CUj"+,

2h h B=l+hp---, C=-. k2

k2

260

Suppose that the number of internal point is N, by above formulas, we get

or

+

Un+l = D(p)Un Rn where:

fBCO...O 0 ABC...O 0

. . . . . .

. . . . . .

. . . . . . 0 0 0 -..BC (0 0 O*..AB By (19),we have

U s = Ds(p)Uo+ D"-l ( p ) R 1+ . . . + D(p)R"-l

+ R"

(20)

Since and can be obtained easily by the expressions of U s and 8D(P) D ( p ) ,the gradient of Q is not difficult t o calculate. 4

numerical result

In this section we show some numerical results. For simplicity, we only consider some one dimensional cases with R = (0, l ) , w = (0.4,0.6). 4.1

Assume that p is a constant

Let ~ 1 ( t = ) exp(4t), r/2(t)= exp(l+4t), and the exact solution u = exp(x+ 4t). From the solution's expression, we know:

$(x, t ) = exp(x + 4t),

&(x) = exp(x + 47)

26 1

The results of using finite difference method and RBF method are given by Table 1 and Table 2 respectively. Table 1.

Table 2.

4.2

Assume that p(x) is a function

Now we investigate the case when p is a function and the measurement with error. Let ~ l ( t= ) exp(6t), r ] 2 ( t ) = exp(1 6 t ) and the exact solution u(x,t)= ezp(x2 6 t ) , we have

+

+

+(z,t)= ezp(z2 + 6 t )

+

$,-(x) = exp(x2 6.r) We put a random error of 1%on the measurement, the regularization is necessary since the illposedness of the inverse problem. By taking qg and pg as the estimations of the q and p with error of 30%, we do the reconstruction.

262

The results of regularized (Figure 3 - 8) and unregularized (Figure 1, 2) are plotted respectively, where 0 represents the result obtained by RBF and * represents the result obtained by finite difference method. Figure 3 - Figure 8 show the results obtained by use of different regularization terms where regularization terms are a11q - qg1I2 PIIp - pg112, allq - qg [I2 + Pllp' (XI11' , and allq - (rs [I2 P C; 1 (xi+l)- (xi)l2 respectively.

;

+

+

U

Figure 2: ,u

Figure 1: q P

Figure 3: q

Figure 4: p

263 P Z8

,

,

,

,

,

,

,

,

,

2s24-

22

-

X

Figure 5: q

Figure 7: q

5

Conclusion

Our research shows that to solve the optimization problem from inverse problem based on solving direct problems by FDM and RBF can get the explicit expression of gradient for the cost function and obtain satisfactory results. However, there are still some questions to be studied, such as the existence, uniqueness and stabilities etc.

264

References

1. Y.C.Hon & Zongmin Wu "A Numerical Computation for Inverse Boundary Determination Problem" 2. Masahiro YAMAMOTO and Jun Zou "Simultaneous reconstruction of the initial temperature and heat radiative coefficient. 3. M.S.Pilant,W.Rundell.An inverse problem for a nonlinear parabolic equation. Commun Partial Differ Equations.1986,11,445-457

NUMERICAL RECONSTRUCTION OF PIECEWISE CONSTANT POTENTIAL FOR ONE DIMENSIONAL HELMHOLTZ EQUATION FUMING MA AND FANGFANG SUN School of Mathematics,Jilin University,Changchun, 130012,P. R. China E-mail: [email protected]. cn In this paper, we study the numerical method of reconstructing potential of one dimensional Helmholtz equation for given impedance function. First,the properties of impedance function with piecewise constant potential for one dimensional Helmholtz Equation was given. Then, numerical method for reconstructing potential was discussed.

1

Problem

Let us consider one dimensional Helmholtz equation as follows: where @(z,k) is a complex function, k wave number and potential q ( z ) real function. Setting we assume that

0 < no I n(.)

5721.

For any complex number k,we are concerned with the solutions 4+(z, k ) and #-(x, k) of equation (1) which are of the form

4+ (z, k ) = 4inc+ (z, k ) + &at+ 4-

(z, k ) = 4272-

( 5 ,k )

+ $scat-

(5,

k),

(2, k ) ,

where

4inc+(z,k ) = e i k x , 4inc-(z,k ) = e P i k x , and 4scat+(x,k), q5scat-(x, k) satisfy with the outgoing radiation boundary condition:

+Wscat*(CJ,k ) = 0

(2)

4Lcat*(Lk ) - ik4scat*(L k) = 0.

(3)

4Lcat*(0,k) and

265

266

Here and in sequel of this paper, we denote by f’(z, k ) the partial derivative for any function f(z,IC). Let

2

C+ = { k E CIIm(k) 2 0). For any k and $+(z, k),define the impedance functions p + ( z , k ) and p - ( z , k ) as follows:

Our problem is: for given impedance function p+(O,k) for all k , reconstruct potential q(z), z E [O, 11. In the case q(z) E Cr[O, 11 and m > 2, Chen and Rokhlin (see Chen and Rokhlin2) proved that impedance functions p + ( z , k ) and p - ( z , k ) are defined well for all z E R and k E C+, and they satisfy with the following Riccati equations: P;@, k ) = - W P : ( z ,

k ) - (1 + Q ( E ) ) ) ,

+

p1_(z,k ) = i k ( p ? ( z , k ) - (1 q ( z ) ) ) .

(6)

(7)

In Chen and Rokhlin2, equations (6) and (7) are used to numerically reconstruct q(z). The numerical results show that, for sufficiently smooth q(z),the method in Chen and Rokhlin2 works very well. But in some cases of problems, potential q(z) is discontinues. In this paper we want to extend the method in Chen and Rokhlin2 to the numerical reconstruction of discontinues

4x1. 2

Discussion on Impedance Functions

Let f(z)be the function on [0,1]. For any division of [0,1]

A : 0 = zo < ~1 < ... < zn = 1, define

267

We denote by T V ( f ) total variation of function defined on [0,1], i.e., T V ( f ) = SUP{V(A)I. A

In this section, we will rewrite impedance function p + ( z , k ) as p ( z ) for reason of simplification. Consider Riccati equation

+

(8)

P ' ( 4 = ik(P2(.) - (1 q ( 2 ) ) ) . we have that

Theorem 1 Assume that p(x) be the solution of equation (8) satisfying with condition p ( 0 ) = po, po > 0, q ( X ) E C[O,l], q(z) > -1 and for z # [0,1], q(z) = 0. Then for TV(ln(1 q ( z ) ) ) < +m, p ( z ) is defined well for z E [0,1] and

+

SUP { I P ( z ) 1 7lP'(z)l) 5

XE[O,11

< fm.

(9)

+

proof: Let ~ ( z = ) 1 q(x). It is easy t o prove that there exist piecewise constant functions {qn(z)}, z E [0,1], n = 0 , 1 , . . ., such that 4n(X)

-+

4(X),

in

L"0,11,

-+

00,

and TV(ln(l+q,(X))) _< k < +m. Denote bypL(z) the solutions of equation (8) with q(z) = qn(z) and p ( 0 ) = po. By using of the method in JSylvester', we can prove that there exists constant M > 0, such that SUP {IPn(z)I, IPL(z)II 5 M . XE[O,'l

Finally,from ArzelB-Ascoli theorem and the uniqueness of solution of initial value problem for ordinary differential equations,we can get the estimate (9). By use of the above theorem, we can prove the following results for impedance functions p + ( ~k,) and p - ( ~k, ) :

Theorem 2 Assume that q(z) be continue on (-m,+m), q(z) > -1, and for x # [0,1], q ( z ) = 0. I n addition, assume that TV(ln(1 q ( z ) ) ) < +oo. Then for all k E (0, +m) and z E [0,1], impedance functions p+(z,k ) and p - ( x , k ) are defined well, and satisfy with equation (6) and (7).

+

Furthermore, we have that

268

Theorem 3 A s s u m e that q ( x ) is piecewise constant function, q(x) > -l,and f o r x @ [ O , l ] , q(x) = 0. T h e n impedance function p + ( z , k ) is defined well f o r all k E (0, +GO) and x 6 [0,1]. Furthermore, p + ( z , k ) is a continue o n x and p + ( z , k ) satisfies equation (6) at point x if x is not discontinue point of q ( x ) .

3 Numerical Reconstruction of Potential q ( x ) In this section, we consider the numerical method to reconstruct potential q(x). Our numerical method is based on the following idea: To find q(x) by solving the system which consists of Riccati equation

P ; ( z , k ) = - q P : ( x , k ) - ( 1 + q ( x ) ) ) , x E [0,11

(10)

and

which is from Chen and Rokhlin2 (Trace theorem),with initial conditions P+(O, k ) = P o ( k ) , Vk

>0

and

q(0) = 0.

By this way, we can get the approximation qh(x) to q(x). After this, we optimize the functional

to get the better approximation of q(x), wherepo(qh, k ) is the impedance function of problem (1)-(3) defined by (4) for q ( x ) = qh(x) and can be obtained by solving (1)-(3) numerically. In the numerical implementation of the above method, we set p l ( x , k ) = Rep+(x, k ) , p z ( x , k ) = Imp+(x, k ) , so that equation ( 1 1 ) can be reformulated as the system

P X Z , k ) = 2P1(2, k)P2(2,k ) ,

(13)

(14) P h k ) = -NP?(Z, k ) - P k k ) - ( 1 + Q(Z))l. For solving numerically this ordinary differential system,we use the following difference scheme

269

where h is difference step-size and xj = jh,# = pl(xj,k ) , j = 0 , 1 , . . . ,M,l = 1,2.This is an explicit and stable scheme. For computing q ( x ) numerically from equation (ll),we can choose a large enough a , and substitute equation (11)by

d then,for 1 = 1 , 2 , .. . , M

-

m=

la

Rep+(x, k)dk,

(17)

1, get

where h = a / N , kj = j h , j = 0,1,. .. ,N , by use of trapezoid formula for integrating (17). By the above method, we did some numerical experiments for piecewise constant function g ( x ) , numerical results of reconstruction for q ( x ) are satisfying.

Acknowledgments This work is partly supported by Special Funds for Major State Basic Research Projects in China (G1999032802) and National Nature Science Foundation of China (Foundation item:10076006).

References 1. JSylvester, A convergent layer stripping algorithm for radially symmetric impedance tomography problem, Comm. in PDE 17, 1955-1994(1992). 2. Y.Chen and V.Rokhlin, On the inverse scattering problem for the Helmholtz equation in one dimension, Inverse problems 8 , 365-391 (1992).

ALGEBRAIC SOLUTION FOR THE INVERSE SOURCE PROBLEM OF THE POISSON EQUATION T. NARA AND S. A N D 0 The University of Tokyo, 7-3-1, Hongo, Bunkyo, Tokyo, 113-0033, JAPAN E-mail: [email protected]

In this paper, a non-iterative, algebraic method for an inverse source problem of the three-dimensional Poisson equation is proposed. The method is based on the multipole expansion of the potential by point sources. Via the multipole expansion coefficients of the sectoral harmonics and the particular tesseral harmonics, the relations between the source parameters and the surface integral of the boundary data are derived. These relations are reduced into an algebraic equation of N th degree for N source positions projected onto the zy-plane. The number of the sources N is obtained by the property of the leading principal minors of the Hankel matrix composed of the multipole expansion coefficients of the sectoral harmonics. Stability of our algorithm is analyzed, and a numerical simulation is shown.

1

Introduction

The inverse source problem of the Poisson equation has many important applications in science and engineering, such as estimation of current sources inside the brain from the electric potential or the magnetic field measured on the head surface. So far, many numerical algorithms for estimation of several spatially localized sources have been proposed. Though the most basic method is the iterative algorithm which minimizes the error between the boundary data and the solutions of the direct problem1 , the direct estimation of the point source parameters by the boundary data has been studied as well for acceleration of algorithms or calculation of the initial values for the iterative algorithms. Ohe et a2 proposed the method to estimate the positions of the point sources in the unit circle in two-dimensional space. Assuming that the source strength is known, they derived the equation of N-th degree whose solutions are the positions of the sources. They also showed an algorithm for the estimation of the number of sources3. Badia et al proposed an explicit algorithm to estimate the source positions as eigen values of a matrix composed of the surface integral of the Cauchy data weighted by harmonic functions. The positions in three-dimensional space, the source strength, and the number of sources with an assumed upper bound can be estimated, though the estimation of the number is unstable as they themselves mentioned. In this paper, we will derive a relation between the source parameters and the surface integral of the Cauchy data via multipole expansion in Sec. 270

27 1

2. The multipole expansion coefficients of the sectoral harmonics and the particular tesseral harmonics yield the relations between the source positions, strength, and the surface integral of the Cauchy data. It is shown that the relations of the sectoral harmonics are equivalent t o the equations used by Badia et aL4. In Sec. 3, we propose another direct algebraic solution: The N source positions projected onto the zy-plane can be represented as the N solutions of the equation of N-th degree whose coefficients can be expressed by the multipole expansion coefficients of the sectoral harmonics. The source strength and the z-coordinates are also expressed by the projected positions and the multipole expansion coefficients of the sectoral and tesseral harmonics. N is obtained by the leading principal minors of the Hankel matrix composed of the multipole expansion coefficients of the sectoral harmonics. In Sec. 4, the stability of our algorithm is analyzed. A numerical simulation is shown in Sec. 5. 2

Relation Between the Source Parameters and the Boundary Data Via Multipole Expansion Coefficients

Let us consider the three dimensional Poisson equation

AV=-f (1) in a bounded domain G E R3,where the source term f is assumed to be the point sources: N

f

=Cqks(T-Tk,e-ek,+-+k), k=l

qk

#o

(k=1,2,...,N).

(2)

Our inverse source problem is to estimate the source strength q k , the source positions r k , o k , &, and the number of sources N, from the Cauchy data

where v is the unit outward normal vector to dG. Let V' be the potential that would exist if the source f were in an infinite medium. Then, it is known5 that in a multipole expansion of V', at a point outside a sphere which contains G, expressed as

272

the expansion coefficients (multipole coefficients) can be represented by both the surface integral of the boundary data and the source term f as anm +ib,m =

l,(va,a

=

av av

(rnPr(cosB)eim@)- -rnPr(cosB)eim@

f rn P r ( c o s 0 ) eim@dv.

(5)

When f is the point sources in Eq. (2), Eq. (5) is reduced t o the basic relation between the surface integral of the boundary data and the source parameters

an,

+ ib,,

=

l,

(V$

av av

( P P r ( c o s O ) e i m @-) -rnPp(cosO)eim@

N

qk'r~~r(COS6k)eim@k

=

) dS (6)

k=l

for n 2 m 2 0. Here, we use the sectoral harmonics component; a , z a,,+ib,, and the tesseral harmonics a t n = m + l ; ,Bm am+~,m+ib,+~,,, for the estimation of the source parameters. Let [

x

+ iy,

Ck

xk

+iyk,

(7)

then by substituting rmP~(cosO)eim@=(2m - 1)!![", r m + l P ~(cosB)eim@=(2m +l + l)!!Cmz, (8) into the Eq. (6), we obtain

The multipole expansion coefficient of n = m in Eq. (9) is equivalent t o the equation used by Badia et aL4. Badia et al. also mentioned the use of the polynomial zQ(x, y), where Q is a harmonic polynomial. The multipole expansion coefficient of n = m 1 in Eq. (10) corresponds to this polynomial, though they used the other Q ( x ,y) in the numerical simulation6. In the next section, we derive another explicit representation of the source parameters. We propose a stable algorithm for the estimation of the number of sources N .

+

273

3

Direct Representation of the Source Parameters

3.1 Explicit Expression of Positions and Strength

It is remarkable that the following linear relation between ai, ai-1, ..., a i - ~ holds for i 2 N :

The estimation of the number N of sources is shown in the next section. Now, let the m x m Hankel matrix composed of (YO to ~ 2 denote ~ ~ 2

=

Hm Hm=

( 8' .; ) i -1, ... ... .. . . ..

a1

ffm-1

a,-2

am-1

-.. ff2m-3

then the following lemma holds.

Lemma 1 m

am-1

a a;,

a2m-2

(14)

274

Let wm,k E (1 0 such that 11 uo IIH1(O,l) ~ .have proposed a new methodology based on a rationality hypothesis for interpreting real world data. The interpretation is carried out by inverse optimization. Inverse optimization is classified into static one and dynamic 0ne10711>12>13. In this paper we focus on the former, which estimates a criterion function under which given data become optimal subject to given constraints. A resulting criterion function provides interpretation of given data. We have proposed a neural networks approach to static inverse optimization for estimating quadratic criterion functions corresponding to given data. A crucial idea here is neural network architecture representing the optimality conditions for both optimization and inverse optimization. Taking advantage of this duality, static inverse optimization problems can be solved by learning of neural networks. This idea alone, however, is not sufficient for solving static inverse optimization. To overcome various difficulties, we have also proposed algorithms for generating constraints from given data, guaranteeing positive semidefiniteness of resulting criterion functions, estimating simple and understandable criterion functions, and interpreting non-Pareto optimal data. Although it can solve static inverse optimization problems and interpret real data, it still has a difficulty in interpreting large-scale real data due to computational complexity in generating constraints. 420

42 1

Generation of constraints requires computation of a convex-hull from given data. Although many algorithms for calculating a convex-hull from a set of points in 2-D and 3-D have been proposed, they are not applicable brute force techniques for to real data in higher d i m e n s i ~ n s l ? ~Existing ?~. calculating a convex-hull from given data are not feasible due to excessive computation time even when the number of given data is fairly smalls. To overcome this difficulty we propose an efficient method for generating constraints by divide-and-conquer. The main features of the proposed,algorithm are the following. It randomly divides large-scale data into subsets, calculates Pareto optimal data for each subset, and calculates Pareto optimal data for the entire data by fusing them. It can be proved that resulting Pareto optimal data are the same as those obtained directly from the original data. By reducing non-Pareto optimal data as much as possible, computational cost for generating constraints becomes much smaller than that by an algorithm without divide-and-conquer. To evaluate the effectiveness of the proposed method, simulation experiments are carried out by using rented housing data (about 4,000 samples) with 4 attributes. They are obtained from tenants living along Yamanote and Soubu-Chuo lines in Tokyog. The proposed divide-and-conquer method requires less than 30 minutes for generating constraints from given data. In contrast a method without divide-and-conquer would require 3,330 years. Section 2 presents formulation of optimality conditions. Section 3 shows a neural network architecture representing the optimality conditions. Section 4 describes a procedure for data interpretation. Section 5 illustrates an efficient method for generating constrants. Section 6 provides interpretation of largescale real data. Section 7 concludes this paper. 2

Optimality conditions for static optimization

We consider the following static optimization with a quadratic criterion function, 1 min f(z)= - z T ~ z sTz 2 2

+

s.t. gz(z) = bra: 5 dz, i = 1,.. . 1 'm (2) where z E Rn is a variable vector, A E !Rnxn is a symmetric positive semidefinite criterion matrix, s E !Rn is a criterion vector, bi E Rn is ith coefficient vector, and di E R1 is ith constant in the constraints. A Lagrangian function, L , is,

+

L ( z ,A) = f(z) ATg(z).

(3)

422

where X is a Lagrangian multiplier vector. The following Kuhn-Tucker condition3 is necessary and sufficient for static optimization.

v s L ( s O A") , =0 g(z0) 5 0,

XOTg(s") = 0

A" L 0 where so is the optimal solution and Ao is the corresponding Lagrangian multiplier vector. Since v s g i ( s o ) = bi and vzf(z")= As" + s, Eq.(4) is rewritten as,

-(Aso

+ s ) = C XPbi.

(7)

i

+

Eq.(7) indicates that a gradient vector of a criterion function, -(As" s), lies inside the polar cone formed by the coefficient vectors { b i } ( i = 1,. . . , q; q 5 m) corresponding to active constraints. Here we assume, without loss of generality, that the first q constraints are active and the rest are inactive . Based on the above formulation, Eqs.(5) ( 6 ) (7), and A 2 0 are the necessary and sufficient conditions for optimality. 3

Neural network architecture

We propose the linear neural network architecture in Figure 1 representing the optimality conditions for static inverse optimizationloill.

1hq L,*%

u bq

b,

-Axe- s -A

v X0

Figure 1. The structure of a neural network representing the optimality conditions

The solution, so,is given to the rightmost block of the input layer in Figure 1. The vector -(As" + s ) is produced at the next layer by propagating the activation through the connection weight matrix, A. -s corresponds to

423

a bias. Therefore, the rightmost module in Figure 1 represents the left-hand side of Eq.(7). Similarly left modules with inputs, bl, . . ., b,, corresponding to active constraints represent the right-hand side of Eq.(7). It is to be noted that both optimization and inverse optimization can be represented by the neural network in Figure 1. In optimization A , s and b l , . . ., b, are given, and A1, . . ., A, and x" are determined by minimizing mean square output error. In inverse optimization 61, . . ., b, and x" are given, and A1, . . ., A,, A and s are obtained by learning of neural networks4. 4

Procedure for data interpretation

A procedure of interpreting data by static inverse optimization is the following.

Step 1 It is assumed, without loss of generality, that the smaller the value of an attribute is, the the more preferable it is. Attribute values are transformed accordingly. Step 2 Generate constraints from given data. During generation, an efficient method for generating constraints is used. Step 3 Select a Pareto optimal sample, and obtain the corresponding active constraints. Step 4 Estimate a criterion function matrix and a Lagrangian multiplier corresponding to the given sample. During learning, a criterion function matrix is modified to guarantee its positive-semidefiniteness. Step 5 After learning by backpropagation, learning toward pseudo-diagonal is carried out for estimating a simple and understandable criterion function. Necessary modification of a criterion function matrix to guarantee positive-semidefiniteness is also done. Step 6 A given sample is interpreted based on the resulting criterion function, lagrangian multiplier, marginal rates of substitution and so forth. Steps 4 and 5 correspond to static inverse optimization, and Step 2 corresponds to the proposed method describled in Section 5.2. 5

5.1

Efficient method for generating constraints

Generation of constraints

In interpreting data, only data are given and constraints are not provided. It is, therefore, necessary to generate constraints from given data for their interpretation. A concept of Pareto optimality popular in welfare economics plays an important role.

424

We assume here, without loss of generality, that the smaller the value of a variable is, the more desirable it is. Under this assumption, x* is Pareto optimal if x satisfying the following inequalities does not exist.

x* 2 2, 3 j x; > xj

(8)

Let the number of data be N and the number of attributes be M . A hyperplane in M-dimensional data space determined by the data {uil,. . . ,uiM}

Figure 2 illustrates the number of hyperplanes, r , as a function of the number of data, N , and the number of attributes, M. Those hyperplanes which satisfy the following two conditions constitute a set of Pareto optimal data. The first condition is that all data exist on one side of a hyperplane and the origin lies on the other side. The second condition is that the sign of all coefficients of the hyperplane are the same. Obtained hyperplanes correspond to partial surfaces of the convex-hull for given data. M.5 M.4 M.3

M.2

f

1000

100

10

1 1

2

Figure 2. The number of hyperplanes, number of attributes, M .

5 T,

10

20

50

100

N

a s a function of the number of data, N , and the

5.2 Procedure for generating constraints We propose the following procedure for generating constraints by divide-andconquer.

Step 2-1 Divide given data randomly into several subsets. Step 2-2 Eliminate non-Pareto optimal data from each subset as much as possible by hyper-ellipsoid and hyperplane elimination algorithms.

425

Step 2-3 Obtain Pareto optimal data in each subset. Step 2-4 Calculate Pareto optimal data for the entire data by fusing them. Step 2-5 Generate constraints from Pareto optimal data. It is proved that the resulting Pareto optimal data are the same as those obtained directly from the original data. [Proof] Let D be a set of original data, Di be ith subset of D , P be a set of Pareto optimal data of D , Pi be a set of Pareto optimal data of Di. D = D1 u D2.. . U D k , P' = PI u P2 u . . .u Pk. Suppose x E Din P , i.e., x is Pareto optimal, this means that y satisfying x 2 y, 3j xj > yj, does not exist in D. It is clear that y E Di,satisfying x 2 y, 3 j xj > yj, does not exist. Accordingly Pi 2 Di n P is satisfied. Taking the union of both sides, we obtain

P ' = P ~ u P 2 ' . ' u P k2 P Accordingly P is included in P', therefore we can obtain Pareto optimal data from PI, and obtain P without loss. [End] 5.3 Two Elimination Algorithms 0

Algorithm of hyper-ellipsoid elimination Calculate the average, fi, and variance and covariance matrix, set of data, U , with N samples and M attributes.

1

O - 2i k

=

N - 1 .C ( U j 2 - / q ( U j k - bk),

2,

k

9, from a

= 1,.. . , M

(11)

3=1

Discard samples with Mahalanobis distance, g,(x,y), smaller than y. TA-1

c

ge(x,y)= (a:- b )

2

:.( - f i ) L Y

(12)

U' is the remaining set of samples with N' samples. Algorithm of hyperplane elimination Find the minimum value of each attribute. gi= uji,

ji

= argminuji, 3

i

=

17 . . . , M

(13)

Determine the hyperplane, gd(x) = 0, composed of M samples, gi(i= i l , . . . , i ~ ) . Discard data satisfying gd(x)> 0. The remaining samples constitute the set U" with N" samples.

426

Application to rented housing data

6

It is assumed that a tenant of a rented house makes a decision by maximizing one's utility. Based on this assumption we interpret real data of rented houses in Tokyog. fa

D-

Hamamawcho

Figure 3. Yamanote and Soubu-Chuo lines in Tokyo

Figure 3 illustrates a map of Yamanote and Soubu-Chuo lines in Tokyo. The number of rented housing data, composed of separate house and apartment houses, along Yamanote and Soubu-Chuo lines is 3932. The attributes of the data are rent, commuting time to Shinjyuku station, area of housing and year of construction. Table 1 provides examples of data near Shinjyuku station. Table 1. Examples of data near Shinjyuku station. y1: rent(104 yen), yz: commuting , y4: year of construction (year). time(min.), y3: area of housing ( m 2 ) and attributes

data Y1

Y2

Y3

Y4

2

5.8 6.5

16 12

14.13 19.15

1982.4 1976.4

86 87

40.0 45.0

13 14

107.6 144.9

1985.1 1989.3

1

Firstly, necessary modifications are made according to Step 1in Section 4. They are the area of housing and years after construction. New variables are: X I = y1, x2 = y2, 5 3 = 287 - y3 and 5 4 = 2002 - 5 4 . Paramenters in

427

these transformations do not directly affect the interpretation, because only the marginal rates of substitution matter in interpretation as will be shown later. Secondly, we generate constraints from modified data according to Steps 2 and 3. Because the number of the data is very large, the proposed method is iteratively carried out, i.e. 8 times, for generating constraints. 19 Pareto optimal data, and the following 21 constraints are obtained.

I

91 :

92 : 921 :

+ +

+ +

+ +

305x1 410.522 65.023 7.424 = 21239 1196x1 621x2 225x3 635x4 = 80537 524x1

+ 588x2 + 107x3+ 2342x4 = 50980

Computation time is 30 minutes due to divide-and-conquer. It would require 3,330 years without divide-and-comquer a. Steps 4 and 5 are omitted here due to space limitation. Table 2. Marginal rates of substitution between attribute Pareto optimal data. T h e 1st column is renumbered.

No.

1’ 2’ 3’ 18’ 4’ 17’ 11’ 5’ 13’ 14’ 19’ 16’ 12’ 9’ 10’ 15’ 7’ 6’ 8’ -

4:;

( 104yen/min.) 0.7-1.9 0.7 0.7 0.1-0.9 0.1-0.7 0.1-2.1 0.5-7.5 1.5-7.5 0.1-1.3 0.9-1.3 0.5-4.4 0.9-1.3 0.9-4.4 3.1-125 1.5-125 0.9-1.3 1.3-7.5 4.6-125 125

4;

( 104yen/m2) 4.7-5.3 4.7 4.7 1.9-32. 1.9-5.3 0.7-5.3 0.7-6.5 4.7-6.5 1.9-32. 18.~32. 0.7-1.0 1.9-18. 1.0-18. 1.0-5.9 0.9-5.9 0.9-32. 3.3-18. 2.1-6.5 2.1

21

(rent) and other attributes for

4

region

( 104yen/year) 1.9-41. 41. 41. 0.1-0.3 0.3-41. 0.1-0.5 0.3-1.9 0.4-1.9 0.1-0.3 0.1-0.2 0.1-0.5 0.1-0.3 0.1-0.3 0.1-0.2 0.1-0.9 0.1~0.3 0.2-1.3 0.1-1.3 0.1

Shinjyuku I1 I1

I1

Shin-okubo Takadanobaba Mejiro Ikebukuro Yoyogi Shibuya I1

Koenji Ogikubo Kichijyoji I1

I1

Musashi-koganei Kunitachi If

Finally, we interpret the rented housing data according to Step 6. Table 2 presents the marginal rates of substitution between attribute 5 1 (rent) and a P C : CPU 1.4GHz, Memory 128MB with Mathematica ver.4.1

428

other attributes. Table 2 suggests that the decision maker 1’ will pay 7,000 19,000 yen to decrease commuting time by 1 munite. The decision maker 1’ will also pay 47,000 53,000 yen to increase the area of house by 1 m2 and will pay 19,000~ 4 1 0 , 0 0 0 yen to renovate a house by one year. Other Pareto optimal data can be interpreted in the same way. The distribution of these Pareto optimal data has three characteristics. The first is that the Pareto optimal data alone Yamanote line are concentrated around Shinjyuku with commuting time of less than 11 minutes. The second is that the Pareto optimal data along Soubu-Chuo line are located west of Koenji. The third is that Pareto optimal data do not exist along Soubu-Chuo line between Sendagaya and Ochanomizu. Table 3 presents the average values of marginal rates of substitution for Pareto optimal data along Yamanote and Soubu-Chuo lines. N

-

Table 3. Average values of attributes and marginal rates of substitution for Pareto optimal data. items rent (lo4 yen), . . commuting time (min.) area of house (m’) year after construct ion (year) pi:; (104 yen/min.) (104 yen/mZ)

p g (104 yen/year)

I

1

I

Yamanote 32.7 11.7 86.8

Soubu-Chuo 23.9 34.1 106.

9.7

2.3

1.59

33.2

8.19

7.50

11.6

0.36

From Table 3 we can say the followings: 0

The longer the commuting time is, the larger the monetary value of commuting time becomes. This is because those tenants who live far from Shinjyuku, have stronger desire to decrease the commuting time. The longer the commuting time is, the smaller the monetary value of years after construction becomes. This is because those tenants who live far from Shinjyuku have weaker desire to live in new houses.

7 Conclusions We have proposed an efficient method for generating constraints by divideand-conquer to interpret large-scale real data from a viewpoint of optimization. It is proved that the resulting Pareto optimal data are the same as those obtained directly from the original data.

429

We have applied the proposed method to largescal real data, and have successfully estimated a criterion function governing decision making of the tenants living along Yamanote and Soubu-Chuo lines in Tokyo. These results well accord with data and our intuition. References

1. D.R. Chand and S.S. Kapur, “An Algorithm for Convex Polytypes,” Journal of the ACM 17-1, 78-86 (1970). 2. A. Datta et al., “A Connectionist Model for Convex-Hull of a Planar Set,” Neural Networks 13, 377-384 (2000). 3. H.W. Kuhn and A.W. Tucker, “Nonlinear Programming,” Proceedings 2nd Berkeley Symposium on Mathematical Statistics and Probability J. Neyman (Ed.) , University of California Press (1951). 4. D.E. Rumelhart et al., “Parallel Distributed Processing,” The MIT Press (1986). 5. J.V. Neuman and 0. Morgenstern, “Theory of Games and Economic Behavior,” John Wiley & Sons, Inc. (1967). 6. H.A. Simon, “The Science of the Artificial,” The MIT Press (1981). 7. E. Wennmyr, “A Convex Hull Algorithm for Neural Networks,” IEEE Trans. on Circuits and Systems 36-11, 64-68 (1989). 8. R. J.-B. Wets and C. Witzgall, “Towards an Algebraic Characterization of Convex Polyhedral Cones,” Number. Math. 12, 134-138 (1968). 9. Recruit Co. , Ltd, “Rented Housing Information [Metropolican Area] ,I1 Ken Corp., Ltd , 2/2 (2000). 10. H. Zhang and M. Ishikawa, “A Neural Networks Approach to Inverse Optimization,” The 2nd R.I.E.C. International Symposium on Design and Architecture of Information Processing Systems Based on the Brain Information Principles (DAIPS) , 197-200 (1998). 11. H. Zhang and M. Ishikawa, “A General Solution to Static Inverse Optimization Problems Using Neural Networks Learning,” The Trans. of IEEJ 120-C(6), 857-864 (2000)(in Japanese). 12. H. Zhang and M. Ishikawa, “A Neural Networks Approach to Dynamic Inverse Optimization Problems,” The Trans. of IEEJ 120-C(4), 481-488 (2000)(in Japanese). 13. H. Zhang and M. Ishikawa, “Structure Determination of a Criterion Function by Dynamic Inverse Optimization,” Proceedings of 7th International Conference on Neural Information Processing (ICONIP-2000) , 662-666 (2000).

This page intentionally left blank

Section V

Related Topics

This page intentionally left blank

EGUCHI-OKI-MATSUMURA EQUATION FOR PHASE SEPARATION: NUMERICALLY GUIDED APPROACH T. HANADA Department of Mathematics, Chiba Institute of Technology, Narashino, Chiba 275-0023, Japan E-mail: [email protected]

H. IMAI Department of Applied Physics and Mathematics, Faculty of Engineering, University of Tokushima, Tokushima 770-8506, Japan E-mail: [email protected] N. ISHIMURA Department of Mathematics, Faculty of Economics, Hitotsubashi University, Kunitachi, Tokyo 186-8601, Japan E-mail: [email protected]. ac.jp M.A. NAKAMURA College of Science and Technology, Nihon University, Kanda-Surugadai, Tokyo 101 -8308, Japan E-mail: [email protected] Eguchi-Oki-Matsumura (EOM) equations are introduced to describe the dynamics of pattern formation which arises from phase separation in some binary alloys. The model extends the well-known Cahn-Hilliard equation. We report our studies of the EOM equation, with an emphasis on numerical analysis.

1 Introduction

This is a report of our recent studies on the Eguchi-Oki-Matsumura (EOM) equation for phase separation with an emphasis on numerical analysis from the inverse problem viewpoint. The dynamics of pattern formation resulting from phase separation has been a fascinating topic for researches. Cahn and Hilliard based on a continuum model in thermodynamics, made a phenomenological approach to explaining such kinetics and derive the fourth-order partial differential equations (PDEs), known as the Cahn-Hilliard equation. Many studies have been performed on this equation and much progress has been achieved so far from various points of view Eguchi, Oki, and Matsumura4, on the other hand, introduced a system of 778,

2,3,5)9310,11112.

433

434

equations, which extends the Cahn-Hilliard equation and consists of coupled two phase fields; one is the local concentration and the other is the local degree of order. After performing a suitable scaling of parameters presented shortly later, EOM equations in one-space dimension, with which we are mainly concerned, are expressed as follows. ut =

+ + v2)u)xx

--E2uxxxx ( ( a

+ (b

~l= t uXx

-

21, = u x x x = 21, Ult=O

= uo,

u2 - V =0

t>O

inO 0 on 0 5 x 5 I ,

~ ) Y

= Yo

&O

inO 0, there exists a unique solution ( u , v ) to (1)such that u E L ~ ( ( o , T ) ; H ~ ( onLm([o,T);H2(o,1)), ,~))

v

E

L ~ ( ( o , TH) ;~ ( o , ~n)L) ~ ( [ T O),;~ ~ I( ) ) .0 ,

For any initial data above, the solution ( u ,v) converges as t 00 t o a solution of the steady state problem (2). (2) has at least one monotone non-trivial steady solution f o r all large b >> m2. Moreover, f o r any integer k 2 2 and f o r all large b >> m2 depending o n k, (2) has at least one non-monotone non-trivial steady solution, each of whose derivatives changes sign exactly (k - 1)-times. --$

2

Existence of Solutions

To establish the local in time existence, a standard Galerkin approximation method is implemented. Let 3 denote the complete orthonormal system in L2(0,Z) with the even periodic boundary condition: 3 := { l / d , m c o s ( ; r r x / l ) , ~ c o s ( 2 ; r r x / l.).,. , m c o s ( n ; r r x / l ) ,. . . }. For every positive integer N , let WN be the linear space spanned by {l/d, m c o s ( n x / l ) ,. . . , m c o s ( N n x / l ) } and PN denote the orthogonal projector in L2(0,1) onto W N.

436

We are then looking for an approximate solution ( u N ( x , t )v,N ( x ,t ) ) to (1) given by

The components { u n ( t )v, n ( t ) }satisfy a system of ordinary differential equations, which has a unique solution on [O,TN)for some TN > 0. Thanks to various a priori estimates, we are able to let N 3 cm;in particular, we have liminfN,, TN 2 T > 0 for some T > 0. Uniform bounds of H1(O,1)-norms enable us to repeat the local solvability procedure and continue the solution. As a summary, our existence results are formulated as follows.We refer to Hanada et a1 for the details.

Proposition 1 Suppose that U O , W O E H1(O,l) with (uo), = (vo), = 0 at 1 x = 0,l and ( l / l ) uo dx = m. Then, for each T > 0 , there exists a unique solution ( u , v ) to (1) such that u E L 2 ( ( 0 , T ) ; H 3 ( 0 , 1nLm([O,T);H1(O,l)), ))

so

w E L ~ ( ( o , TH) ;~ ( o , ~n )L,([o, ) T ) ;H ~ ( o , L ) ) . Concerning the long time behavior of the solution (u,v) to (l),rather routine inference involving the Lyapunov functional F [ u ,v] works, and we conclude that ( u , v ) tends to an element of the w-limit set of (uo,vo),on which F [ u ,v] is constant; namely, ( u ,v) converges to an equilibrium solution of the steady state problem (2).

3

A Priori Estimates

We here collect some a priori estimates, which is needed to prove the existence and to determine the asymptotic profile of the solution ( u ,v) to (1). To start with, we introduce the following function spaces.

ET := { ( u ,V) E L 2 ( ( 0 ,T); H4(0,1 ) ) x L 2 ( ( 0 T); , H 2 ( 0 ,1)) 1 u, = uxZz = v, = 0 at x = O , l } ,

EO:= {(uO,vo)E (H2(0,1))2~ ( U O ) ,= (vo), = 0 at x = O , l ,

437

where T > 0. The norm 11 . 1) denotes that of L2(0,1). Furthermore, COstand for various constants depending only on the initial data and constants E ~ a ,, 6, which may differ from line t o line. We understand that COis independent of

t. Lemma 1 There holds Ilv(t)ll,

I max{Ilvoll,,

h}

f o r 0 < t < T.

I n particular, Ilv(t)ll, I & f o r all large t. Lemma 2 For any initial data verijies

(210,

vo) E Eo, the solution ( u ,v) E ET to (1)

f o r 0 < t < T , and moreover

Ilu(t)ll, I co. Lemma 3 I t follows that f o r any 0

It I s 5 T

Lemma 4 There holds f o r any 0 5 t 5 s 5 T

The proof of above lemmas are combinations of integration by parts and the application of Gronwall's inequality. The computations are tedious but straightforward; the detailed expositions are found in Hanada et a1 8 .

438

4

Structure of Steady Solutions

The structure of steady state solutions to EOM equations, that is, solutions u = u ( z ) and v = ~ ( xwhich ) verify (2) is investigated. Our results read as follows, which extends our previous establishments '.

Proposition 2 For all large b >> m2, there exists at least one monotone nontrivial steady solution for EOM equations. Furthermore, for any integer k 2 2 and for all large b >> m2 depending o n k, EOM equations have non-monotone non-trivial steady solutions, each of whose derivatives changes sign exactly (k - 1)-times. We remark that the large values of b and m2 stated in Proposition 2 can be computed explicitly. 5

Computational Study

Our numerical scheme is motivated in part by that for the Cahn-Hilliard equation '. Let xk = k A x (k = 0 , l , . . . , n ) with Ax = l / n . The discretized free energy P [ U ,V] for the approximations ( u k , v k ) of ( U ( X k , t ) , V ( x k , t ) ) is expressed as

1 + a-U: 2 + -V: 4

b

- -V; 2

+ Z1 U ~ V ~ ) A X .

Here V+ and V- denote the forward and backward difference in x , respectively:

C represents the trapezoidal summation formula defined by =

1

+

cuk"+ -us.

N-1

1

2

k=l

Now, for the approximations ( o k , v k ) of ( u ( x k ,t mainly adopt the implicit scheme as follows:

+ At),V ( x k ,t + A t ) ) ,we

439

where V 2 := V+V- stands for the second order central difference in x. With this implicit scheme, we deduce that the discretized free energy is decreasing6:

P [ U ,V ] I P [ U ,V ] . This property is useful if the method is applied to the computation of inverse problems. Here we supplement our solver by the explicit scheme, since it is fast and a posteriori stable to implement.

In this case, the dissipation of the free energy holds only approximately. The discretized boundary conditions should be fixed as

U-1 = U1, Un-l = Un+l v-1 = v1, Vn-1 = Vn+1 u-2 = Uz, Un-2 = Un+2

in place of u, = 0 at x = 0 and 1 in place of v, = 0 at x = 0 and 1 in place of uzZz= 0 at x = 0 and 1.

We focus our interest on the question whether exists the variety of steady state solutions or not; taking constants 1=1,

&=l,

1

u=m=4’

several steady solutions are now illustrated in following Figures. Figure 1 depicts the convergence of a solution (u, v) for (1) to a monotone steady solution. We set b = 16/25 and as initial function we employ

440

vo(x) =

Jcz-

1

cos(7rs).

The computation is implemented under the mesh size 1/256 up to the time interval 0 5 t 5 4096.

Ax

=

1/64 and

At

=

V

U

1.25

1.00

0.15

0.75 0.50 0.25

0.50 0.25

. t

0.00

t

X

X

Figure 1. Convergence to a monotone steady solution.

The monotone steady solution corresponds to the case k = 1 in Theorem 1. It is numerically unstable with respect to the perturbation on initial data. Figure 2, on the other hand, illustrates the convergence to a non-monotone steady solution. We set b = 0.99, and as initial function we take ~ o ( z= ) m,

The implementation data are the same as those of Figure 1, while we perform during the time interval 0 5 t 5 128. The limiting function is related to the case k = 2 in Theorem 1; however, the function v is monotone increasing in Figure 2. This apparent discrepancy is easily reconciled by virtue that the sign of v is irrespective to the problem. We hasten to remark that the non-monotone steady solution, which is constructed in Theorem 1 with k = 2, also is numerically realized. Finally, we exhibit the energy diagram of various steady solutions in Figure 3. Here the notation 20 - i (i = 0,1,. . . ,8) means the steady solution to (l),which is akin to the one with k = i 1 described in Theorem 1.

+

441 V

U

1 0.75

0.5 0.25 0

t

X

Figure 2. Convergence to a non-monotone steady solution.

0.00

-0.05 -0.10

-0.15 -0.20

0.5

1.0

1.5 b

Figure 3. Energy diagram of steady solutions.

Acknowledgments We are grateful to Professor Hiroshi Fujita for his interest in this research. Thanks are also due to the referee for various comments, which helps improving the manuscript. This work is partially supported by Grants-in-Aids for Scientific Research (Nos.10555023, 12640223, 13555021, 13640206), from the Japan Ministry of Education, Science, Sports and Culture.

442

References 1. J.W. Cahn and J.E. Hilliard, Free energy of a nonuniform system, I., Interfacial free energy, J. Chem. Phys. 28, 258(1958). 2. J . Carr, M.E. Gurtin, and M. Slemrod, Structured phase transitions o n a finite internal, Arch. Rational Mech. Anal. 86, 317-351(1984). 3. C.M. Elliott and D.A. French, Numerical studies of the Cahn-Hilliard equation f o r phase separation, IMA J . Appl. Math. 38,97-128 (1987). 4. T. Eguchi, K.Oki, and S. Matsumura, Kinetics of ordering with phase separation, Mat. Res. SOC.Symp. Proc. 21, Elsevier, 589-594(1984). 5. C.M. Elliott and Zheng S., O n the Cahn-Hilliard equation, Arch. Rational Mech. Anal. 96, 339-357 (1986). 6. D. Furihata and M. Mori, A stable finite difference scheme f o r the CahnHilliard equation based o n a Lyapunov functional, Z. angew. Math. Mech. 76, S1, 405-406(1996). 7. T. Hanada, N. Ishimura, and M.A. Nakamura, Note on steady solutions of the Eguchi-Oki-Matsumura equation, Proc. Japan Acad. Ser.A. 76, 146(2000). 8. T. Hanada, N. Ishimura, and M.A. Nakamura, O n the Eguchi-OkiMatsumura equation f o r phase separation in one-space dimension, preprint, (2001), submited. 9. T. Hanada, M.A. Nakamura, and C. Shima, O n Eguchi-Oki-Matsumura equations, GAKUTO Int. Ser. Math. Sci. Appl. 12, 213(1999). 10. A. Novick-Cohen, Energy methods f o r the Cahn-Hilliard equation, Quart. Appl. Math. 46(4), 681-690(1988). 11. A. Novick-Cohen and L.A. Segel, Nonlinear aspects of the Cahn-Hilliard equation, Physica D 10(3), 277-298(1984). 12. S: Zheng , Asymptotic behavior of solution to the Cahn-Hilliard equation, Applicable Anal. 23, 165(1986).

SOME RESULTS ON THE EXACT BOUNDARY CONTROLLABILITY FOR QUASILINEAR HYPERBOLIC SYSTEMS TA-TSIEN LI Department of Mathematics, Fudan University Shanghai 200433, China E-mail: dqliafudan. edu.cn BOPENG RAO Institut de Recherche Mathe'matique Avance'e Universite' Louis Pasteur d e Strasbourg , Strasbourg, France In this paper we present some results on the local exact boundary controllability for general one-dimensional first order quasilinear hyperbolic systems with general nonlinear boundary conditions and give corresponding applications to nonlinear vibrating string equations.

1

Introduction

First of all we recall the definition of exact boundary controllability for hyperbolic equations (systems). For a given hyperbolic equation (system), for any given initial data 'p and final data +, if we can find a time TO> 0 and suitable boundary input controls on the boundary dR of the domain 52, such that the corresponding mixed initial-boundary value problem with the initial data 'p admits a unique classical solution u = u ( t , z ) on the whole domain [0, 7'01 x which verifies exactly the final condition

a,

t = To: u = + ( 2 ) ,

2 E

R,

(1)

namely, if by means of boundary input controls the system can drive any given initial state 'p to any given final state at t = TO,then, we say that this system possesses the exact boundary controllability. More precisely, if the exact boundary controllability can be realized only for initial and final states small enough in a certain sense, we say that the system possesses the local exact boundary controllability; Otherwise, we say the system possesses the global exact boundary controllability. There are a number of publications concerning the exact controllability for linear hyperbolic equations (systems) (see J. L. Lions', D. L. Russell' etc.). For the nonlinear case, using the HUM method suggested by J. L. Lions and Schauder's fixed point theorem, E.Zuazua3 proved the global (resp. local)

+

443

444

exact boundary controllability for semilinear wave equations in the asymptotically linear case (resp. the super-linear case with suitable growth conditions). Furthermore, using a global inversion theorem, Lasiecka and Triggiani4 established an abstract result on the exact controllability for semilinear equations. As applications, they gave the global exact boundary controllability for wave and plate equations in the asymptotically linear case. However, only a few results are known for quasilinear hyperbolic systems. In one-dimensional case, the exact boundary controllability for reducible quasilinear hyperbolic systems was proved in Li-Zhang5 and Li-Rao-Jin by a constructive method which does not work in the general case of quasilinear hyperbolic systems. In an earlier work, M. Cirin&879considered the zero exact boundary controllability for general quasilinear hyperbolic systems with linear boundary controls, but the author needed some very strong conditions on the coefficients of the system(global1y bounded and globally Lipschitz continuous). Moreover, if one applies the result of M. CirinA8 twice t o get the general exact boundary controllability, the corresponding controllability time should be doubled. In this paper, we will present some results on the local exact boundary controllability for general one-dimensional quasilinear hyperbolic systems with general nonlinear boundary conditions and give corresponding applications to nonlinear vibrating string problems. 617

2

General Considerations

Since the hyperbolic wave has a finite speed of propagation, the exact boundary controllability of a hyperbolic equation (system) requires that the controllability time TOmust be suitably large. In order to have a classical solution t o the corresponding mixed initial-boundary value problem on the domain [0, TO]x we should first prove the existence and uniqueness of the semiglobal classical solution, namely, the classical solution on the time interval 0 5 t 5 TO,where TO > 0 is a preassigned and possibly quite large number. The exact boundary controllability will be based on the existence and uniqueness of semi-global classical solution t o the mixed initial-boundary value problem of quasilinear hyperbolic equations (systems). On the other hand, in order to realize the exact boundary controllability, it is only necessary t o find a time To > 0 such that the given hyperbolic equation (system) admits a classical solution u = u ( t , x ) on the domain [0, TO]x 0, which verifies simultaneously the initial condition

a,

t=O:

u=cp(x), z € R

(2)

and the final condition (1). In fact, putting u = u ( t , x ) into the boundary

445

conditions, we get immediately the boundary controls. By uniqueness, the classical solution to the corresponding mixed initial-boundary value problem with the initial data cp must be u = u ( t ,x ) , which automatically satisfies the given final data $. Moreover, if the solution u = u ( t , x ) constructed in the previous paragraph also satisfies a part of boundary conditions, then we need only to put u = u ( t ,x) into the other part of boundary conditions to get the corresponding boundary controls, and, as a result, the number of boundary controls will be reduced and the boundary controls can be asked to act only on a part of boundaries, however, the controllability time will be enlarged. Of course, for the purpose of application, the controllability time To will be asked to be as small as possible.

3

Main Results

We now consider the following first order quasilinear hyperbolic system

where u = ( 2 1 1 , . . . ,u , ) ~is a vector valued function of ( t ,x ) , A(u) = ( a i j ( u ) ) is a n x n matrix with suitably smooth elements aij(u) ( i , j = l , . . . , n ) , F : R" + R" is a vector valued function with suitably smooth components fi(u)(i= l , - . . , n ) and

F ( 0 ) = 0.

(4)

By t,he definition of hyperbolicity, for any given u on the domain under consideration, the matrix A(u) has n real eigenvalues Xi(u)(i = 1, . . . ,n) and a complete set of left eigenvectors & ( u )= (Zil ( u ) ,. . . ,Zin(u))(i = 1,. . . ,n):

4 ( u )A(u ) = Xi ( u )li ( u ),

(5)

and, correspondingly, a complete set of right eigenvectors ri(u) = (Ti1 ( u ) ,. . . ,r&))T (i = 1,.. .,72):

A ( u ) T ~ (= u )X~(U)T~(U).

(6)

We have (resp. det lrij(u)I # 0).

det Ilij(u)l # 0

(7)

Without loss of generality, we may assume that li(U)Tj(U) 2

sij

( i , j = 1,.. . , n )

(8)

446

and T

ri (u)ri(u)3 1

(i = 1 , . . . , n ) ,

(9)

where S i j stands for the Kronecker symbol. Moreover, in this paper we assume that on the domain under consideration, the eigenvalues satisfy the following conditions:

Let

We consider the following mixed initial-boundary value problem for the quasilinear hyperbolic system (3) with the initial condition

and the boundary conditions

x=O: x =1:

u ',

V,

=G,(t,vl,...,v,)+H,(t) = G,(t,v,+l,...,V,)

+H,(t)

(s=m+l,-..,n), (T

= l,...,m).

(13) (14)

Without loss of generality, we assume that

Gi(t,O,.-.,O)= O

( i = I,... In).

(15)

For a preassigned and possibly quite large number TO> 0, we have the following existence and uniqueness of semi-global C1 solution u = u ( t ,x ) to the mixed initial-boundary value problem (3) and (12)-(14) (See Li et a1 lo and Li et a1 "). Theorem 1 Assume that lij(u), Xi(u), f i ( u ) , Gi(t,.), H i ( t ) ( i , j = 1 , . . . , n) and p ( x ) are all C1 functions with respect to their arguments. Assume furthermore that (4), (7), (10) and (15) hold. Assume finally that the conditions of C1 compatibility are satisfied at points (0,O) and (0,l) respectively. Then, for a given TO> 0, the mixed initial-boundary value problem (3) and(l2)-(14) admits a unique C1 solution u = u(t,x ) (called the semi-global C1 solution) with suficiently small C1 norm on the domain

W o ) = {(t,X)l 0 L t

i To,

0 Ix

i I},

(16)

provided that the C1 norms ~ ~ $ ~ ~ and ~ ~ ~[ ~o ,Hl [~ ~ are c Ismall [ ~ enough , ~ ~ ~(depending on TO).

447

Based on Theorem 1, we can get the following theorem on the local exact boundary controllability (See Li et a1 and Li et a1 1 2 ) .

''

Theorem 2 Assume that lij(u), Xi(u), f i ( u ) and Gi(t,.) ( i , j = l , . . . , n ) are all C1 functions with respect to their arguments. Assume furthermore that (417 (7), (10) and (15) hold. Let

For any given initial data cp E C1[0,1] and finial data $ E C1[O,11 with small C' norm, the quasilinear hyperbolic system (3) admits a C1 solution u = u ( t , x ) with small C1 norm on the domain R(To), such that

t=O:

u=cp(x), o < x < 1

and t=To:

u=$(x),

O < X < ~ .

Therefore, we can find boundary input controls Hi E C1[0, TO] (i = 1,.. . ,n ) with small C1 norm, such that the mixed initial-boundary value problem (3) and (l2)-(l4) admits a unique C1 solution u = u ( t ,x ) on the domain R(To), which verifies the final condition

t=To:

u=$(x),

O_-

L

drn’

Then, for any given initial data cp E C2[0,11, $ E C1[O,11 and final data @ E C2[0,1], 9 E C’[O,l]withsmall C1 norms $ ) ~ ~ c I [ and ~,~I 9 ) 1 1 ~ 1 [ ~and , ~ 1for any given function h(t) E C2[0,T]with small C’ norm ~ ~ h ’ ~ ~ satisfying c ~ [ ~ , ~the] ,following conditions of C2 compatibility at points (0,O)and ( T ,0 ) respectively: h(O) = cp(O), h‘(0) = $(O), h”(0) = K’(cp‘“))cp’’(O) + F(cp’(O), $Cl(O))

(37)

h ( T ) = @(O), h’(T) = 9 ( 0 ) , h”(T)= K’(W(O”W’(0) F(@’(O),*(O)),

(38)

and

+

there exists a boundary control h(t) E C2[0,T ] with small C’ norm IIE’IICI[~,~I in case (31) or h ( t ) E C1[O,T]with small c1norm Ilhllcl[o,T] in cases (32)(34), such that the mixed initial-boundary value problem for equation (27) with the initial condition t = 0 : u = cp(x), U t = $ ( x ) , (39) the boundary condition (30) a t the end x = 0 and one of the boundary conditions (31)-(34) at the end x = 1 admits a unique C2 solution u = u ( t , x ) on the domain

which verifies the final condition

Acknowledgments The author Ta-tsien Li was supported by the Special Funds for Major State Basic Research Projects of China.

451

References

1. J. L. Lions, Contr6labilit.4 Exacte, Perturbations et Stabilisation de Systbmes Distribue's ( Vol. I, Masson, 1988). 2. D. L. Russell, Controllability and stabilizability theory for linear partial differential equations, Recent progress and open questions, SIAM Rev. 20, 639-739( 1978). 3. E. Zuazua, Exact controllability for the semilinear wave equation, J. Math. Pures et Appl. 69, 1-32(1990). 4. I. Lasiecka and R. Triggiani, Exact controllability of semilinear abstract systems with applications t o waves and plates boundary control problems, Appl. Math. Optim. 23, 109-154 (1991). 5 . Ta-tsien Li and Bing-yu Zhang, Global exact boundary controllability of a class of quasilinear hyperbolic systems, J. Math. Anal. Appl. 225, 289-31 1(1998). 6. Ta-tsien Li, Bopeng Rao and Yi Jin, Solution C1 semi-globale et contr6labilit.4 exacte frontibre de systbmes hyperboliques quasi line'aires re'ductibles, C. R. Acad. Sci., Paris, t.330, Skrie I, 205-210(2000). 7. Ta-tsien Li, Bopeng Rao and Yi Jin, Semi-global C1 solution and exact boundary controllability f o r reducible quasilinear hyperbolic systems, M2AN 34,399-408(2000). 8. M. CirinA, Boundary controllability of nonlinear hyperbolic systems, SIAM J . Control Optim. 7, 198-212(1969). 9. M. CirinA, Nonlinear hyperbolic problems with solutions o n preassigned sets, Michigan Math. J. 17, 193-209(1970). 10. Ta-tsien Li, Yi Jin, Semi-global C' solution t o the mixed initial-boundary value problem for quasilinear hyperbolic systems, Chin. Ann. of Math. 22B, 325-336(2001). 11. and Yi Jin, Solution C1 semi-globale et contr6labilite' exacte frontibre de systbmes hyperboliques quasi line'aires, C. R. Acad. Sci. Paris, t.333, Skrie I , 219-224(2001). 12. Ta-tsien Li and Bopeng Rao, Exact boundary controllability for quasilinear hyperbolic systems, SIAM J. Control Optim, Submited. 13. Ta-tsien Li and Bopeng Rao, Local exact boundary controllability for a class of quasilinear hyperbolic systems, to appear in Chin. Ann. of Math., 23B (2002).

This page intentionally left blank

Author Index AmmariH. 3 AndoS. 270 Anikonov D.S. 13 BanksH.T. 26 BaoG. 37 Chan Y-H. 325 Chang M.M.Y. 394 ChenQ. 143 ChengJ. 225 Choy Sh-0. 325 DahlkeS. 56 DingG-H. 403 Eskin G. 105 Filipowicz S.F. 336 FuCh-L. 237 FungY-H. 325 GizaZ. 336 HanB. 356 HanW. 349 HanadaT. 433 H0nY.C. 291 Huang S-X. 349 Imai H. 247,433 IsakovV. 47 Ishikawa M. 420 Ishimura N. 433 JiaCh-X. 255 JiaX-Zh. 225 Kawashita M. 182 Kawashita W. 182 KimS. 114

Konovalova D.D 13 Kovtanuyk A.E. 13 LesnicD. 123 LiG-Sh. 143 Lip-J. 411 LiT-T. 443 LiuG.R. 314 LiuH-F. 374 Liu J-J. 134 LiuJ-Q. 356 LiuK-A. 356 MaF-M. 265 May-Ch. 143 MaaP Peter 56 Matsumcto T. 384 Mig6rski S. 160 Nakamura G. 192 Nakamura M.A. 433 NaraT. 270 Nazarene V.G. 13 Neubauer A. 67 NgM.K. 364 Ohtsubo H. 314 0rS.H. 394 Prothorax I.V. 13 QiuCh-Y. 237 Ralston J. 105 RaoB-P. 443 Romanovski M.R. 171 Sabatier P.C. 84 Semoushin I.V. 28 1 453

454 Ship-Ch. 374 Sikora J. 336 SiuW-Ch 325 SogaH. 182 SunF-F. 265 Takeuchi T. 247 TanY-J. 255 TanakaM. 384 TanumaK. 192 Trooshin I. 202 WangX-L. 356 Wang Y-B. 225 WangZ-W. 301 WeiT. 291

W0ngK.H. 394 XUD-H. 301 XuY.G. 314 Yamamoto M. 114,202 Yamamura H. 384 YaoW. 403 Y e s . 212 Ying G-J. 143 ZengY-B. 212 Zhang G-Q. 41 1 ZhangH. 420 ZhaoH-B. 356 Zhu Y-B. 237

This page intentionally left blank